Copernicus’ model didn’t remove the need for epicycles etc.: To get results fully consistent with the most precise observations of the day, his model needed about as much complication as the Ptolemaic model did. What really set it apart wasn’t the precise matches: It was the fact that even the simplest version still gave a fairly good approximation, which the Ptolemaic model didn’t do at all. Copernicus’ model gets you retrograde motion, planets being brightest when they’re retrograde, planets being in opposition when they’re retrograde (which the Ptolemaic model never really explained at all), and the planets which don’t go retrograde always being near the Sun, all without need for any messiness at all.
Copericus’s model was just the beginning of the heliocentric model, not its endpoint. He had to have epicycles because he still had circular orbits. The Tychonic model had the same. It wasn’t until Kepler that elliptical orbits were introduced.
If we’re comparing fully developed geocentric models vs. fully developed heliocentric models, we may not be able to tell the difference with just naked-eye observations. But there are differences that can be seen with telescopes. Since the geocentric model didn’t have gravitational influences between planets, there would be movements of the planets that it would not be able to account for. (I don’t think they could add epicycles to allow for perturbations, but I could be wrong.) As I said above, Neptune was predicted based on the gravitational influence of an unseen body with Uranus. That couldn’t happen with the geocentric model.
Now I suppose someone could develop a geocentric model that had gravity. That might or might not be indistinguishable. As far as I know, no one has done so.
Again, it’s been done, with the results 100% indistinguishable from ordinary general relativity. It’s a horrendous mess, but it works.
It’s difficult to see how going from a heliocentric model to a geocentric model is not a simple transformation. We do this all the time by inserting so-called ‘fictitious forces’ such as the Coriolis force into Newtonian physics and such forces can be treated as additional fields along side the ‘physical’ fields such as gravity. What we don’t usually do is, for example, work out the Coriolis force acting on Neptune due to the assumption of a geocentric frame as that frame is not usually the easiest to do calculations when we need to include both the Earth and Neptune, but there is nothing intrinsically difficult about doing so. Some might not be happy with the frame-dependent terms that have to be added to Newton’s laws by the insertion of fictitious forces, but Newtonian physics can be reformulated to have laws that are invariant under general coordinate transformations. General relativity was formulated from the start in a way that is invariant under general coordinate transformations so can equally handle a geocentric frame, the problem being that you will probably come up against the lack of procedure to extend an arbitrary local frame in GR.