Either my understanding of orbital mechanics is completely wrong or that is completely incorrect. Geostationary satellites need very regular station keeping otherwise they either fall to Earth or are ejected out into solar orbit. If ejected it could remain operational for a while but if it fell back to earth the results would be obvious.
Your understanding of orbital mechanics is totally wrong. Geostationary satellites do need frequent stationkeeping maneuvers, but that is because the satellite is required to remain in a 30km box. If these maneuvers cease, as would happen with the sudden disappearance of humans, they will start to drift off their stations, eventually collecting in a couple of regions, one over the Indian ocean and the other over the Pacific. (This is due to the earth's slightly uneven gravity). Because of the vastness of space, the probability of them actually running into each other is fairly low.
A geostationary satellite would need almost the same amount of energy to come down as it takes to put it up there, and probably twice as much to escape the earth's gravity well. At the end of life of these satellites, they use the remaining fuel to boost them another 200km or so in altitude, then vent all remaining fuel (so they won't explode if there's a fuel leak), and then blow the electronics to make sure they don't interfere with anything else. They will remain in that graveyard orbit forever.