2. I'm fairly certain it was entirely due to POP3 access to my Gmail account. By default this is enabled. I didn't use POP3 access, so I've since disabled POP3 on my gmail. Everybody!!!! DISABLE POP3 ON YOUR GMAIL NOW.
Not if you weren't using it. The POP3 implementation would have to have a serious security hole that allowed bruteforcing a password or something like that. You can't send through POP3 either. The only normal way would be if you used POP3 (which you say you didn't) over a non-secure connection and someone/something was snooping at the time.