Slitherine

I notice that there is a shipyard next to a city I conquered, and it still belongs to the other nation. I cannot build anything there, and it shows with his flag in the lower right of the screen.

1) Can he still build ships there?
2) If so, where does the population come from to build those ships?
3) And, if not, is he still paying support for this shipyard?

Similar questions would apply to other structures. It appears that the AI nations are keeping structures which no longer seem to benefit them (or maybe they do? but how?), yet it seems to me that in this game, paying upkeep on such structures is a rather significant matter.

Hi Beorn,
thank you for the questions.
You take over shipyard once the initial owner doesnt own any tiles around the shipyard. If this is not the case (a.k.a. you already own all the tiles around shipyard and it is still not yours), it is a bug.

Ad1) Basically yes, if there are his cities nearby (up to certain distance). If not, he can still build them, but it will cause revolts in his cities (people are withdrawn from far away cities and they dont like it).
Ad2) Pretty much answered in Ad1 - from cities further away, causing revolts.
Ad3) Yes

AI will eventually destroy/sell those buildings. Till he decides to do so, he still pays upkeep costs.