Whatever is going on, this is not a virus.
Regarding the incoming e-mails, have you checked your junk mailbox? It's possible that, through error or design, most of those messages have ended up in there.
Since you're having other problems with Mail, though, that's certainly not the only possibility. Is Mail the only app that is misbehaving? If so, you may want to try reinstalling the latest combo update:
You could also try resetting Mail completely. First, quit Mail, then hold down the option key and choose Library from the Finder's Go menu. In that folder, find the Mail folder and move it to the desktop. Next, go to the Preferences folder (also in the Library folder) and move the com.apple.mail.plist file to the desktop. Now re-open Mail. It will be as if you have opened it for the first time. Set up all your accounts from scratch, then import any mailboxes that you need from the Mail folder on your desktop. If anything in this process goes wrong, just put those two items back where they came from, replacing any newer items that have been created in their place.