How precise can an artificial satellite be when it comes to following the exact orbit in relation to Earth or hitting the same point above Earth in orbit? Which orbit would be the most stable for this?
It depends upon the orbit and what time scales you are talking about. Satellites are subjected to many perturbations in its orbit. There are effects due to atmospheric drag, which as you'd expect affect lower satellite orbits more than higher orbits, but the atmosphere swells up all the time depending upon the level of solar activity. Gravitationally, the Earth is not a point mass and it has regions where the gravity gradient changes, which causes the satellite to get pulled one way or another (very slightly) as it orbits around. There are a host of other small, but not insignificant perturbations to the orbit, so predicting satellite orbits is a lot like predicting weather forecasts. You start with its known state (position and velocity) and you propagate the orbit to make a prediction. The model you'd use to make your prediction would include things like atmospheric drag to attempt to get a better prediction, but just like weather models, all those little perturbations and errors add up over time, so your orbit prediction gets worse the further out you look in time.
The most precisely known orbits in use today are for earth-observing science satellites that are part of the DORIS (Doppler Orbitography by Radiopositioning Integrated on Satellite) system, such as Jason and CryoSat, using ground-broadcast Doppler radio for range-rate measurement, laser ranging, and GPS simultaneously. Their orbits are known to about one centimeter in the radial direction, and about three centimeters in the other two directions. This precision is necessary to accomplish their first mission, which is to use radar to measure the sea surface height, which varies very slightly with temperature (since water expands as it warms, though not by much!)
In the case of the three Jason missions (launched 2001, 2008, and 2014) and their direct ancestor, TOPEX/Poseidon (launched 1992), all four spacecraft are in nearly identical orbits, because that so-called "frozen orbit" (the wikipedia article is terrible, but it very well illustrates how complicated it is to even attempt such a thing) was chosen extremely carefully in order to minimize the natural drift over time caused by the fine details of Earth's nonspherical gravity and the effects of the sun, moon, and other planets. Even the first of them was so stable in its orbit that they maintained the semimajor axis to within 7 meters (one part per million) for the first four years using a total of only nine maneuvers, which to me is astonishing. It is also why Jason-1 is still flying in formation with its successors, despite having ceased active operations a decade ago.