I completed my first Apocalypse mode game recently (marathon mode, huge maps, small continents). The game actually ended before the end game stuff could be triggered (the AI won a diplomatic victory sometime between stage 6 & 7 of the climate change), so I was forced to go the "one more turn" route just to see the fireworks. Thought it was pretty cool when the game map got dark and the comets started to hit. After around 9 comets or so (and a few solar flares) the map's coloring turned back to normal, though the comets kept coming (with solar flares every now and then). Strangely enough, after some time had passed, the map once again got dark... as for the other disasters, as noted, flooding and volcanoes seem to die off once the comets start falling, though every now and then I would see a volcano go off or a river flooding. After about a hundred comets there were still a large number of cities remaining on the map, so I stopped. Conclusions: if you want to get to the end game stuff in Apocalypse Mode maybe consider turning off the diplomatic victory conditions, and also maybe use a smaller map, so that the impact of the comets is more consequential.
Other things I noted: in the first 3 eras there were hardly any World Wonders constructed at all (entering the Renaissance, only 8 wonders had been constructed, 2 of them mine), which made me wonder if the AI was spending more time rebuilding from disasters than focusing on wonders... but then in the Renaissance suddenly 12 Wonders were created, so... seemed to be less missionaries out and about as well. One thing I did note was that almost none of the AIs had sizable armies (Gran Columbia, for example, went through almost all the game with a military strength of 0). Again, I figure the AI was spending so much time repairing damaged districts or building builders that they just didn't have much resources to devote to military units.