Interesting thoughts for sure.
1st answer: See the MASQUERADE_AS directive.
2nd answer: I wouldn't use POP or IMAP for that purpose, I'd use either an alias or set up a procmail rule that fowards the mail for the user to another machine.
3rd answer: No, it's not a trivial problem. In fact, the only solution that I've ever found is to pick a machine that is the authoritative machine for my mail and use IMAP to read the mail from all my other machines. Trying to synchronize mail spools across machines, while possible, isn't something that I'd relish doing.