The main problem with seeing very distant objects is that they're very dim. So little light reaches us from them that we have to record data for days and days to make them distinguishable from the background noise. This is what has been done in the XDF.
It's not commonly appreciated just how many galaxies there are in the visible universe. The number of stars in a big galaxy like the Milky Way is a few hundred billion. The exact figure depends on how many really dim stars you count. This is about the same as the number of large galaxies in the observable universe, and if you count small galaxies as well the figure is several trillion. Again the exact figure depends on how many dwarf galaxies you count.
The point is that if you look at any patch of sky you'll have more galaxies in the field of view than stars from the Milky Way, and it's not hard to find a patch of sky where there are no stars from the Milky Way - just galaxies. This is what the XDF did.
The distant galaxies in the XDF are too faint to measure a precise red shift, but the XDF measures the light over a range of wavelengths so it can get a measure of the galaxy's colour. This gives us an approximate red shift, and from this the distance can be calculated. You might object that galaxies may contain stars of different colours, but the red shifts are much greater than the range of colours within galaxies. In fact we can't see any more distant galaxies because their light has shifted too far into the infra-red for Hubble to see it. The Extremely Extreme Deep Field study will have to wait for the James Webb telescope to be launched.
You ask about focussing and the movement of the Earth, but the Hubble telescope is designed to compensate for this. It can also measure angles to stars with exquisite accuracy, so it's no problem to take pictures from ten years ago and combine them with pictures taken yesterday. Well, I say "no problem", I suspect the Hubble team would say it was a great deal of work! The point is it can be done.
I feel like this answer is a bit unfocussed, but then so was your question. Maybe this will help you refine your question to make it a bit more specific.