What is the relative time difference between us and a star system in outer layer of our galaxy?

Your third question has a very well known answer here:

What means, that although a big time difference had to exist, there is a much smaller one, and in the opposite direction. This is considered the effect of the dark matter.

Our Sun is around 30000 ly from the galactic center, the edge of the galaxy is around 50000 ly, so we can read on the diagram, that the actual speed difference is around 15 km/s, which is 1/20000 of the speed of light. This causes a (not really big) time dilation due to the special relativity.

There is another source of the time dilation, which is caused by the gravity of the galaxy, which differs in the case of us and in the case of somebody living on the outer edge. On such weak gravitation (i.e. for from any black hole, neutron star, etc) it practically depends on the newtonian gravitational acceleration.

But this acceleration is very small - it takes a half galactic year (some hundred millions of year) to revert the orbital speed of the Sun around the galactic core! So, we can consider that actually negligible.

Thus only the special relativistic effect remains. 15 km/s is around 1/20000 of the speed of the light. In cases of speed which are much smaller as the light speed, we can use the speed dilation formula 1/(2*(v/c)^2). Substituting 1/20000 into this formula, we get 1:800 000 000 .

For laymans, we can say, it takes around 25 years for the clocks of this people living on the outer edge of our galaxy to dilate 1 second.