I'd like to look up my achievements without having to boot the game every time I want to look for a guide.
Is there a place online to see what achievements I've got or not?
I'm not aware of any, but in addition to the community site cloudymusic mentioned, Blizzard will provide an API to get all kinds of data about your account (including the achievements) and your heroes.
Not at the moment, but Blizzard is working on a Diablo 3 community site (think WoW's Armory), which will be launching "soon after launch". No concrete ETA that I've seen yet, though.