By my experience, they have issues starting the wars they should be starting. I had Huns next door and they did bring in two customary invasions as expected, yes, but I haven't seen any meaningful attempt by the AI to stop my cultural or diplomatic victories through military force.
Yeah, pretty much. There's probably a buggy / inappropriate check just before they DoW. It happened in some Civ 4 versions too.
I also agree the AI should try stopping the guy who's about to win more, at least when they can.
If somebody is close from domination, ally armies against him.
If somebody is close from diplo win, buy city states or attack him.
If somebody is close from cultural win, up your culture or bring down his tourism (breaking modifiers like ideologies / open borders / trade and so on...), or attack him
If somebody is close from science win, bomb its capital, attack...
AI doesn't do that enough in my opinion... They should react earlier and assess victory status earlier and better.