untagged - How do astronomers choose their research topic?

Let me answer what I think you're asking (parts of your question aren't that clear to me).

Background: I am currently an astrophysics graduate student.

First, from you question it seems to me that you are proceeding under an assumption that you are accepted into a research group when you are accepted into a PhD program. Everywhere I looked as I applied for graduate school, though, this certainly was not the case. Not only was I not accepted into a specific research group at the schools I was accepted to but I was also encouraged to try out different research projects before settling on a specific topic to pursue for my dissertation.

That being said, I do think there are some programs which may choose to accept or not accept you based on what you are thinking you would like to do for research (if you know). This may be because of resources (for instances, if all the professors who are experts in what you want to do research in already have as many graduate students as they want/have money for, that may impact your acceptance and will impact your chances of doing the research you want should you be accepted).

Can you pick and choose?

A specific answer to your question is going to vary significantly from school to school. For example, some programs are very accommodating in letting students "pick and choose" what there want to do research in partly because there are currently resources available to allow that. As explained above, because of resources or other considerations some programs may not have the luxury of opening all doors for its graduate students.

And could you choose whether to do the physical theory or the observations or coding simulations?

Again, this is very school-specific. Some schools have immense computing resources and getting computer time to run simulations/heavy calculations is easy. Other schools may have no computing resources available (other than your desktop workstation) and it will be more difficult to get the computing time you'll need for simulations since you will have to look outside your university for resources. Some schools have access to a lot of time on some very big telescopes; some have access to telescopes that are specialized for specific topics in astrophysics; and other schools may have no access to telescopes at all, meaning you will have to find telescopes with allocation time you can apply for without having to belong to an institution affiliated with that telescope. It varies from school to school.

So, your ability to pursue specific research topics in astrophysics can be enhanced or hampered by the computing and telescope resources available at your department. Take that into consideration as you apply to schools.

SUMMARY

The answers to your questions vary greatly from department to department and are largely dependent on the resources available to a specific department. I have found generally that there is great flexibility in astrophysics programs to choose a PhD dissertation topic but that certain research topics have more support and resources at some institutions than at others. Also, at many institutions professors are limited in how many students they can advise by the amount of funding they have for PhD students, which can significantly affect the research options available to you. There are, however, some institutions that have the resources to cover for an under-funded professor if a student does want to work for them, but again it just boils down to it varies from school to school.