the sun - Declination and Ascension - the Sun and Andromeda

If you mean solar time midnight than the answer is simple (as commented by @rgettman). The Sun should then be at the opposite side of the sky at $alpha = 13^{textrm{h}}$, where $alpha$ is the right ascension. There is a problem, however, when you use local time (using a timezone). If, for instance, you mean midnight GMT in Liverpool, which is at 3° west then solar midnight will happen at 12 minutes past midnight GMT and the Sun would have $alpha approx 13^{textrm{h}}12^prime$.

galaxy - Have we observed any rogue/wandering stars?

Do we know (have we observed and cataloged) any rogue star, being not part of galaxy, but drifting somewhere in inter-galactic space?

I know that determining if a star is a part of galaxy or not is a matter of definition, but let's take those, that inter-galactic space is where the gravitational influence of any galaxy isn't dominating.

the moon - How bright is the full Earth during the lunar midnight?

Let's take an average albedo for the Earth of 0.3 (it depends, which hemisphere is visible, how much cloud cover etc.). That means the Earth reflects 30% of the light incident upon it.

The flux $f$ falling on the Earth is given by
$$ f_{odot} = frac{L_{odot}}{4pi d^2} = 1.369times10^{3} Wm^{-2}$$
where $L_{odot}=3.85times10^{26} W$ from the Sun and $d= 1$AU.

The integrated luminosity from the illuminated hemisphere will be
$$L_{earth} = 0.3pi R^2 f = 5.2times10^{16} W$$

So now we can compare this with the Sun. One hemisphere of the Sun radiates $1.93times10^{26} W$, and produces a flux of $1.369times10^{3} Wm^{-2}$ at 1 AU. Therefore the illuminated hemisphere of the Earth results in a flux of approximately $f_E=0.056 Wm^{-2}$, assuming the average Earth-Moon distance of 384,400 km. This calculation assumes isotropic emission, but it is quite likely that the albedo is higher for light reflected through 180 degrees.

The Sun has an apparent magnitude of -26.74, so the magnitude of the "full-earth" at the moon is
$$ m_{Earth} = 2.5log_{10}left( frac{f_{odot}}{f_{E}}right) - 26.74 = underline{-15.77}$$

The answer will of course vary with the albedo of the visible hemisphere, which in turn depends on the time of year and how much of the polar regions can be seen (e.g. http://www.climatedata.info/Forcing/Forcing/albedo.html ). Variations of a few hundredths seem possible, which will lead to apparent magnitude variations in $m_{Earth}$ of $sim pm 0.1-0.2$ mag. The albedo may also vary in detail with the exact angle at which the sunlight hits the Earth - an "opposition surge" in brightness, when the Sun-Earth and Moon are almost aligned is possible. The Earth-Moon distance varies fro 363,000 to 405,000 km. This will lead to magnitude variations of $pm 0.12$ mag.

A further way to check this is that the albedo of the Moon is 0.12 and it has a radius of 0.273 ties that of the Earth. Therefore the Earth from brightness of the moon ought to be $(0.3/0.12)times(1/0.273)^2 = 33.5$ times brighter. This is 3.81 magnitudes brighter. The mean magnitude of the full Moon is -12.74 (maximum is -12.92), so the brightness of the "full Earth" should be -16.55 on average.

I am not sure why these figures don't agree; I suspect it is that the albedo for reflection when the Sun's light is normally incident on the moon is quite a bit larger than 0.12. The so-called "opposition surge". If the Earth's albedo behaves in the same way, then the latter figure may be more accurate than my first calculation. My gut instinct is that the answer is somewhere between the two.

space - How long will it take to reach the closest earth like planet

HD 219134b is apparently closest "super earth"; 21 light years away.
New Horizons, our speediest yet, is currently running at about 15 km/sec relative to the sun.
At 9.46e12 km/lightyear HD 219134b is 2e14 km away
At 3.16e7 seconds per year, New Horizons will travel about 4.74e8 km/year.
2e14/4.74e8 = 422000 years.

HD 219134b isn't even a very good earth; gravity is too high.

photography - What causes horizontal and vertical lines coming out of pictures of stars?

sometimes you can get these spike artifacts from the microlenses over the CCD's sensor array as well.

also if your CCD does not have antiblooming logic, a very bright star can cause neighboring pixels to saturate as they are being read out, leading to a "spike" only along the readout axis. here's a good document with some common artifacts, from the hubble team:
HandoutIIIc.

the spikes in the OP's image are likely caused by the spider vanes which hold the secondary mirror in place, as Russell says.

observation - Sensitivity of calculated orbital elements to observational errors

These days, we have some very precise ways of making measurements, but I'm sure it wasn't so in Kepler's day. So I am wondering how astronomers of that time could make such accurate determinations of planetary orbits, given the likely limitations of the available instrumentation. To put the question another way: How does each added (or lost) digit of precision in the measurement affect the accuracy of the calculated orbital elements? Is there significant compounding of errors due to the nature of the calculations, or is it essentially one-for-one? How much accuracy (to how many digits of precision) must the "three observations" be in order to determine the orbital elements to a given precision? How does that precision degrade over time? If, for example, we wanted to prepare an almanac for Neptune that will be accurate to say, one second of arc in our sky after 100 years?

observation - Pictures of a curious astronomical phenomenon

In your friends picture are more artifacts than the one you showed in the 2nd picture a little bigger. I marked more of them in the picture below.

They are all in a perfect line to the bright light. So these artifacts are caused by the bright light and the lens of the camera. A lens is not flat it's, well lenticular (it's where the name came from). The main light source is shining from an angel on the lens. When light hits a surface two things happens (at least in most cases ignoring Brewster's angle). The two things are reflection and transmission (or absorption, but we are talking about a lens so i let this one drop here). Everytime the light hits the surface of the lens a part is reflected and an other part is transmittet. But the lens has a thickness so when the light that was first transmittet now hits the back if the lens a part is transmittet to the CCD-Sensor of your Camera and a (smaler) part is reflected back into the lens. This can happen severel times. Normaly this effect is small enouth it will not be noticed. But this picture is not a normal case. Most of the picture (dark sky) emitts no light, the picture is very dark. In kontrast to this there is a brigth light source domenating everything (related to luminescence). And whe have a 7 sec exposure the picture is taken with. 7 sec are enougth time for the CCD-Sensore to capure some light from the effect descriped above.

If you now want to know why is it green and know white. White light includes all wave lengths (colors) and every wave length is reflected in an other angel on the surface of air-lens. The green wave length just had the best angel to be reflected severel time and still reach the CCD-Sensor.

So the most accurate answer to your question „what is this green thing“ is: It's the flashlight (or what ever this light source is) that is seen in the bottom of the picture.

galaxy cluster - How many galaxies are there in the Hercules–Corona Borealis Great Wall?

I'm sorry for the long answer. The result is in the bottom. :)

The number of galaxies in the "wall", as in any volume of space, is the number density $n_{mathrm{gal}}$ times its volume $V_{mathrm{wall}}$. However, the number density of galaxies depends on their masses/sizes: Small galaxies are much more common than large galaxies. Usually we describe this by the halo mass function (HMF), which gives the (differential) number density dN/dM of dark matter halos hosting galaxies as a function of halo mass M. Depending on your favorite cosmology, it looks something like this (created with this tool):

The solid line shows the HMF today in the local Universe, i.e. at redshift zero, while the dashed and dotted line shows how it was at redshift 1.6 and 2.1, respectively, roughly 10 billion years ago, which mark the beginning and end of the wall (according to Horváth et al. 2014, A&A, 561, L12). Notice how the number of large halos grow with time, at the expense of small halos. This is due to the smaller ones accreting onto larger ones.

The HMFs are characterized by a power law at low masses with an exponential cut-off at high masses. To get the total number density we need to integrate over the entire range of masses, and here comes the problem: Whereas the upper limit is set by the maximum size a galaxy can have (roughly $3times10^{13}M_odot$, but the exact number isn't very important, as these galaxies are so rare), the lower limit depends on your definition of a galaxy. Large galaxies have smaller galaxies orbiting as satellites, and there isn't a well-defined threshold for when a clump of gas, stars, and dark matter is called a galaxy. These guys report on the detection of a $sim10^3M_odot$ clump which they call a galaxy. That's probably not what you think of as a galaxy, so for the sake of this exercise, let's use Small Magellanic Cloud-sized galaxies as our lower minimum, i.e $sim 6.5times10^9 M_odot$.

The total number density of galaxies is thus found by integrating the HMF. Luckily, the online HMF tool can do that as well. The result, i.e. the number density of halos above a given mass, looks like this:

Note that the $x$ axis is given in terms of $h^{-1}M_odot$. Since $h simeq 0.7$, the mass of the SMC is $M_mathrm{SMC} simeq 5times10^9 h^{-1}M_odot$. From the graph, the number density of these objects is $sim0.5,h^{-3}mathrm{Mpc}^{-3} = 0.17,mathrm{Mpc}^{-3}$ (1 Mpc, or megaparsec, equals 3.26 million lightyears). In principle we should subtract the number density of halos above $3times10^{13}M_odot$, but since these are four orders of magnitude smaller, they can be neglected.

The wall starts at redshift ~1.6 and ends at redshift ~2.1. This translates into starting at a distance of 4.65 Gpc (billion parsec) and ending at 5.47 Gpc, i.e. a depth of ~800 Mpc. These numbers are in comoving coordinates, i.e. what it has expanded to today, as opposed to physical coordinates, i.e. what it was at the time it emitted the light that we see today. But the HMF is given in terms of comoving volume, so that's cool.

To get the volume, we also need the area. The finders report that it's roughly 2 Gpc times 3 Gpc, but they don't seem to state whether this is in physical or comoving coordinates (which differ by a factor (1+z) ~ 3). I suspect it's comoving, though.

Thus, the total volume is $V_{mathrm{wall}} = 0.8 times 2 times 3,mathrm{Gpc}^3 = 4.8,mathrm{Gpc}^3 = 4.8times10^9,mathrm{Mpc}^3$, or 166 cubic gigalightyears. Here, I have ignored the fact that the physical area spanned by the angle of observation is a bit different at the front and the back of the wall, but since these numbers are already quite uncertain, it's of lesser importance.

The considerations above are all for the average Universe. But the wall represents an overdensity in space, so the number will be larger. I can't find any estimates of the overdensity of this particular supercluster, but in general galaxies in superclusters are more numerous by a factor of $delta sim 5$–$10$. Let's use 5.

That is, the total number of galaxies the size of the Small Magellanic Cloud or above is
$$N_mathrm{gal,tot} = n_mathrm{gal} times V_mathrm{wall} times delta$$
$$= 0.17,mathrm{Mpc}^{-3} times 4.8times10^9,mathrm{Mpc}^3 times 5$$
$$sim 4 mathrm{,billion,galaxies}.$$

But if you go through the same exercise for halos of masses above, say, $10^3,M_odot$, you'll find a number which is a million times larger.

In reality, the number density of luminous galaxies is smaller. At the high-mass end, the gas has trouble cooling sufficiently to condense and create stars, and active galactic nuclei in massive galaxies heat the gas. At the low-mass end, gas is more easily blown out of the potential well by stellar feedback from supernovae.

kepler - Unbelievably high masses/ densities in nasa website data

In May, I did my own amateur analysis of Kepler data that was supplied on Kepler.nasa.gov/Mission/discoveries. This was a table of all confirmed "Kepler planets" to date (April 1, 2015). This table had some unbelievably high masses/densities listed in the first column after the planet designation (ie: planets of Kepler 23,24,25,27,28,32,39,48,52,54,57,58,59,60 and others). I didn't take these seriously, but would like to know if there was a "units" error or something I am missing. To give you an idea, Kepler 23b is given a radius of 1.9 R(Earth) and a mass of 254.3 M(Earth) and that's nothing compared to some others. I checked just now and the data is the same.

Relationship between temperature of nebula and size of star

This is such a complex issue that I'm not going to try to be comprehensive. One major thing you are missing is that stars tend not to form in isolation, especially massive stars, but in groups and clusters.

The basic unit in star formation is the Jeans mass.
$$ M_J propto T^{3/2} rho^{-1/2}$$
where $T$ is the temperature, $rho$ is the density and $M_J$ is the minimum mass for a cloud to gravitationally collapse.

So from this point of view, a higher temperature cloud must be more massive in order to collapse in the first place.

But as the cloud collapses, the density goes up and the Jeans mass decreases - and the cloud fragments into smaller sub-units and ultimately into a cluster of stars.

Exactly how this pans out in practice is basically the whole research field of star formation. It depends not only on relatively simple things like the temperature of the gas, its density, the equation of state; but also on much more complicated ideas like magnetic field support, competitive accretion between stars, feedback from newly-formed stars and protostellar outflows and the rate at which turbulence is generated and then dissipated.

There almost certainly is no simple relationship as you propose. In fact the outcome of star formation seems to produce a mass function (number of star per unit mass) that is only weakly dependent on the initial conditions, but may differ a bit in low density and high density environments.

cosmology - Is the Astronomy community still concerned about the lumpyness of matter distribution in the universe?

The issue is twofold: (a) whether the formation of cosmological structure is an open question and (b) if that is the case who is working on it? Structure formation always was and still is an important issue for cosmology since we need to understand that structure in order to say that we understand our Universe. As with all aspects of science, the nature of the study of structure formation has changed considerably over the past decades: today's problems are quite different from yesterday's. For one thing, we now realise that the baryonic component of the Universe is but a few percent of the total mass density. Over 97% of our Universe is made up of dark matter and dark energy. The game has changed.

(a) The issue of cosmological structure formation goes back to the work of Gamow in the 1940's, but we now have very strong, I would say almost incontrovertible, evidence that the formation of the structure on large scales was driven by gravity. We understand the formation of what is referred to as the "Cosmic web", the large scale filamentary structures surrounding great voids that dominate the observed structure. This is the environment for the formation of the galaxies that we see. We have models for the process of galaxy formation. These are based around numerical gravitatinal N-Body models dressed up with semi-analytic descriptions of the star formation process. The results can look rather good:
http://www.illustris-project.org/
and we can choose parameters that bring such models into some kind of accord with what is observed.

The models tell us, for example, how we might study the dark matter component through observations of satellite galaxies of larger galaxies. Since we can observe the galaxy formation process out to redshifts of 2 or more (the most distant galaxy observed has a redshift in excess of 8) we can check the evolution seen in the models. If you are a cynic you might say we can "re-tune" the models.

(b) We are now measuring the 6-7 parameters that describe our universe by observing various aspects of that large scale structure. We can for example look at the clustering on scale > 100Mpc and detect an excess of clustering at what is called the Baryonic Acoustic Oscillation (BAO) scale, a feature that emerges from the primordial Universe. We can also look at the voids in the cosmic web to do the same thing in an entirely different way. All that can be calibrated by N-Body models and compared with the answers obtained from the cosmic microwave background. So we find groups with nice names like WiggleZ looking at the BAOs:
http://wigglez.swin.edu.au/site/
who use redshift surveys containing millions of galaxies. It's quite amazing, even to one who works in this field.

What's the importance of all this? We find that the Universe can be described by just 6 parameters. We can measure these parameters with any of these methods with an accuracy of a few percent or better. What is convincing us that we have a good model is that these quite independent ways of measuring the 6 parameters all come up with the same answer to within the experimental accuracy. This is "Precision Cosmology". We can feel confident that the Universe is made up of ~70% of dark energy because we detect its influence in different ways and all those ways provide the same values. What is it? That's the 21st century question.

In short - the important questions are still open and hundreds, if not thousands, of astronomers are working on it.

Does the cosmic microwave background recede at the speed of light?

Well obviously the CMBR arrives (towards us, not receding) at us at the speed of light as it is a form of electromagnetic radiation and local velocities are not affected by expansion.

What I guess you may mean though is the surface of last scattering, which is the surface from which the CMBR that we receive at a given instant was emitted in the early Universe. This surface of course is no longer emitting CMBR in the present cosmological time and it is not a fixed (comoving) surface either as which surface we receive the CMBR from changes with time, however we can give a recession velocity to the comoving set of points which make up the surface of last scattering at any given instant.

In the standard model of cosmology, the Lambda-CDM model, the instantaneous surface of last scattering is receding from us at approximately 3.2c (i.e. a recession velocity 3.2 times the speed of light).

What are iron stars? - Astronomy

The basic premise of an iron star is a 'dead' star whose mass is made primarily of iron. It's a hypothetical type of compact star. Wikipedia has a bit of a blurg here: https://en.wikipedia.org/wiki/Iron_star

The process of forming an iron star would be tricky, because normally iron does not fuse and so the star would normally collapse under it's own weight - this is why the Wiki article describes a cold-fusion process via quantum tunneling processes.

This article describes the process of fusion in stars and might help you understand why this is more significant compared to regular stars: http://www.eso.org/public/usa/news/eso0129/

asteroids - Classification of a Comet

Google's definition of a comet is:

a celestial object consisting of a nucleus of ice and dust and, when
near the sun, a ‘tail’ of gas and dust particles pointing away from
the sun.

This doesn't cut it for me. Neither does wikipedia's similar entry.

It pertains to the sun, even though comets are present in other parts of the galaxy other than the solar system and probably other galaxies.

Whats more dwarf planet can be classified as follows:

 * orbits a host star
 * rounded by its own gravity
 * not massive enough to induce thermonuclear fusion
 * not massive enough to clear its region of debris

Which pretty much fits the bill of a comet.

Do you think the following is a fair / recognised classification of a comet:

Comet:
* not rounded by its own gravity
* not massive enough to induce thermonuclear fusion
* not massive enough to clear its region of debris
* composition of ice and dust

But then that's pretty similar to an asteroid:

Asteroid:
* not rounded by its own gravity
* not massive enough to induce thermonuclear fusion
* not massive enough to clear its region of debris

However, this means if an asteroid is comprised of ice and dust, it becomes a comet?

Or are asteroid never comprised of ice and dust? Does my asteroid classification need beefing up, regarding an asteroid's composition?

I'm feeling the comet and the asteroid both need some classification pertaining to their orbit as well? Though thats tricky because only some comets and asteroid, so they're similar in that regard as well! Maybe more details on each objsts composition is needed, like I say...

the moon - What are the Gamma rays and Cosmic rays effects on humans and equipments?

First of all *not to consider me a conspiracy theorist(, but isn't landing on the moon a questionable issue?

Only to conspiracy theorists. To everyone else, no, it's not a questionable issue. My father in law helped send men to the Moon. I have worked with a number of people who sent men to the Moon. I was once called on the carpet in Gene Kranz's office. I find it extremely insulting to think that it is a questionable issue. So excuse me if my response might seem be a bit insulting.

How would the Gamma rays and cosmic rays affect the equipments on the lunar surface and would these equipments function normally like on earth?

Equipment and humans did not suffer immediate damage given the short period of time that humans did spent on the Moon. One of the effects of mild radiation is increased risk of cancer (but that's a long-term effect). The men who went to the Moon (and they did go to the Moon) did indeed suffer increased cancer rates compared to the Earth-bound population.

Could the enormous amounts of solar energy bursts be avoided somehow for protecting human flesh?

By luck, there were no large solar energy bursts when men were in space on the way to the Moon, on the Moon, or coming back from the Moon. A large coronal mass ejection event did happen on August 7, 1972, but that was (by luck) sandwiched between the Apollo 16 and Apollo 17 lunar missions.

How is it possible for an FM or any kind of data transmission type be achieved in an environment that has nothing but an empty space? How would the electromagnetic waves travel and enter the earth's atmosphere to be captured by the receivers?

This question makes no sense. Look up in the sky during the day. What do you see? You see the Sun. Look up in the sky at night. What do you see? You see the Moon, the planets, stars, and if you live in an area with low humidity and limited light pollution, you even see other galaxies. With your naked eye. The only difference between light and FM is frequency. Both are forms of electromagnetic radiation. Electromagnetic radiation travels unfettered through empty space.

What is the bright lunar surface real temperature and is there a way to equip an astronaut to protect him from the over-heat?

The first defense against the temperature extremes of the space environment is very simple: It's coloration. A spacesuit that was jet black in the visible range but white in the thermal infrared would have quickly killed the NASA astronauts on the Moon due to overheating. On the flip side, a spacesuit that was white in the visible range but jet black in the thermal infrared would have resulted in overcooling. The space suits worn by the NASA astronauts on the Moon were white in the visible range but grayish in the thermal infrared. NASA spent a lot (a whole lot) of money investigating different fabrics, different dyes, and different paints. Passive thermal control is the first step in any space operation against the extremes of space. Active thermal control addresses what little passive thermal control can't address.

planet - Did the Late Heavy Bombardment Period Happen Because of a Stellar System Collision?

Recent studies and simulations reported in the paper Impact-induced compositional variations on Mercury (Edgard et al. 2014), suggest that the Late Heavy Bombardment (LHB) was

triggered by giant planet migration during which the main belt and E-belt are excited to higher impact velocities when resonances with the giant planets sweep across the primordial asteroid belt. These simulations indicate the E-belt is the principal source of LHB impactors.

The 'E-belt' is a simulated and theorised primordial extension of the asteroid belt that extended

the primordial asteroid belt could have originally
extended well into the Mars-crossing zone

In the paper The Primordial Excitation and Clearing of the Asteroid Belt (Petit et al. 2001), state an important point to be considered when answering this question:

most of the asteroids
we see today are not primordial, but fragments of larger asteroids
destroyed in a collision. Only the largest asteroids retain characteristics
that relate to the formation of the asteroid belt and were
not drastically changed by the later evolution.

In their simulations, they suggest that the asteroid belt could have been a major feature of the entire inner solar system where the 'planetary embryos are forming (from 0.5 to 4AU), stating:

the presence
of large embryos in the inner Solar System for about 100 to
200 My after Jupiter has reached its present-day mass provides
an efficient mechanism for depleting the asteroid belt of most of
its primordial mass and for dynamically exciting the remaining
small bodies.

Related to the Late Heavy Bombardment:

few percent of the particles end up on these long-lived orbits,
equivalent to several times the mass of the present asteroid belt.
These orbits are unstable on a long timescale and represent a
potential source of impactors for the Late Heavy Bombardment

orbit - State of the stars

Stars rotate due to the angular momentum of the gas they formed from. This angular momentum must be conserved, and remains as the rotation of the star and it's satellites. If a star collapsed from a completely static gas cloud it would not rotate, but would still be stable. The stability is provided by the hydrostatic equilibrium between radiation and thermal pressure with gravitation collapsing the star - not the stars rotation.

I don't know any statistics but I expect all stars rotate to some degree; It is usually the turbulence of gas clouds that leads to overdensities that then collapse in to stars.

Which planet is also the name of an element?

As per ReNiSh A R's post, the answer is Mercury.

A quick Google search took me here:
http://en.wikipedia.org/wiki/Timeline_of_chemical_element_discoveries <= second result!

If you control-F search the page, you'll find the following:

Mercury - discovered before 2000BC, so the planet will have been named after the element

Plutonium - About 1940/41, meaning it would have been named after Pluto (before it was declassified as a planet)

Uranus - Uranium discovered around 1789

Neptune - Neptunium discovered 1940

In addition, the only one of those having exactly the same name for both the planet and the element is Mercury.

Is there light on the surface of Venus?

The best way to answer this question is from images taken from probes that landed and took pictures from the surface of the planet. According to the Lunar and Planetary Institute, images from the Russian Venera 13 lander revealed

The effect of sunlight filtering through the dense atmosphere appears to give the surface an orange tint.

Despite some uncertainty of the actual colour due to possible chromatic distortions because of the atmospheric composition etc. It is clear in the image from Venera 13 (below) that there is indeed light on the surface f Venus.

Image attribution (from the link above): Images courtesy of James Head (Brown University), in collaboration with USSR (now Russian) Academy of Sciences.

Observations, reported in the NASA article NASA Scientist Confirms Light Show on Venus reveal another potential source of light - lightning, and theorised to be

maybe even more activity than there is here on Earth

How to estimate age of asteroid family Erigon

I'm not a professional astronomer, but I'm going to take a stab at this. If anyone sees any problems, feel free to correct them.

The Yarkovsky effect basically means that spinning asteroids over time will be pushed into a further orbit (more precisely, it increases the semimajor axis of the orbit) due to uneven heating and cooling.

Apparently to estimate the age of an asteroid family, using only the Yarkovsky effect, we would start out with this equation:

$$0.2H = log(Delta a / C)$$

This equation assumes a fixed geometric albedo for the whole family. $H$ here is the absolute magnitude. $Delta a$ is equal to $a - a_c$ where $a$ is the semimajor axis.

$$C = sqrt{p_v} (da/dt)_0 T cos epsilon$$

$p_v$ is the geometric albedo. $(da/dt)_0$ is the maximum Yarkovsky drift rate for a body of size $D_0$, where $D_0$ is an arbitrary reference size (we could use your provided $D$ value above). $T$ is the age of the family that we are trying to calculate. $epsilon$ is the spin axis obliquity.

Note: I've had to remove my example solution because I could not figure out how to correct my errors from accidentally putting $C$ outside the logarithm.

You can check out more details in my source below.

Source:

"Yarkovsky/YORP chronology of asteroid families" - Vokrouhlickýa, Broža, Bottke, Nesvorný, Morbidelli

distances - Do objects look larger the further away they are, beyond z=1?

Yes this is correct for the standard model. I believe the English language explanation would be that the angular size of a galaxy does not depend on its current distance to us, but its distance when the light we currently see from it was emitted (here using "distance" synonymous with "proper distance"). Though I also believe there are other complicating factors such as the spatial geometry of the Universe.

Look at the first diagram in Figure 1 in this paper, notice how the size of the past light cone in terms of proper distance firstly increases as we go back in time, before reaching a maximum and decreasing to zero at t = 0.