I think this is a good idea if implemented properly. A #trivia channel with a bot in it would be kind of fun imo. I think the main problem would be who would write up all the Q and A's, then who would code it. (Though Echo seems like he could take care of this part?)
I think a database with everyone whose participated so far's number of correctly answered questions would be a good idea. (Ex. After each correct answer, the bot is like 'Correct-blah-blah, Hito has moved up to rank #4 in trivia, with 63 correct answers to date. etc.)
I don't think exp should be awarded for Total Level/etc though. osu! trivia is osu! triva, osu! is osu!, best to keep them separate.
I'm a little worried about spam, but maybe I'm overestimating how many people would be interested in participating. Also, there's the language barrier problem come to think of it. I'd feel a little bad if all the questions were in English, since if you don't speak English, you're missing out on a feature. I might be worrying for nothing there though.
Well, either way it'd need to be run by peppy, to see what he thinks and if he'd make a #trivia channel and allow it. orz
I think it's a cute idea.