The rules describe that when a barbarian enters a capital city, the owner of that city loses two trade tokens. Does the barbarian token then remain inside the capital city until moved again? If so, is there any other impact to the barbarian being present in the city? (e.g. Are other players required to attack the barbarian before attacking the city? Can caravans enter the city? What if they have the "Currency" upgrade? Is there any impact on world wonders in such a city?)

This isn't addressed anywhere in the rules. I thought that when barbarians enter a space with a capital, they take two trade tokens and go back to where they were, that's how we played it. But I saw that there is no mention of this anywhere in the rule book, so I guess my group and I made it up

I was wondering the same thing in our 1st game this weekend

I think the resolution would be the same as if they had attacked a reinforced control token, with the obvious exception being that you don't flip your capital over. You lose two trade tokens and the barbarian token is removed and returned to the pool awaiting the next star icon on the event dial. It would simulate your government bribing the foreign hordes to prevent continued invasion. At least that's how our group interpreted the rules. Also if you have the ancient wonder Petri, barbarians can't even touch any of your cities. But that's the only wonder I can think of that affects barbarians. Currency is another way of thematically bribing barbarians away from you instead of attacking them for the trade token. Barbarians can occupy City-States, and currency could be used to "liberate" them from barbarians, but I seriously doubt they can occupy your capital.

Yeah we had the same issue and we simply did as above, the Barbarians wouldn't enter the space and remain instead where they were. Next time they move hopefully they'd go somewhere else, or at least it gives the player a chance to defeat them.