1. If Outlook 2003 isn't working too well, you should get that problem sorted out before upgrading.
2. It shouldn't cause any problems, but I always prefer to have all apps at the same level.
3. You can sync between two computers, but not easily and not without buying additional software, assuming you are using POP3. If you really need to do this, you should move your mail account to a hosted exchange-like service, or start using IMAP.