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.