[Brian]

I seem to remember us getting an email from someone experiencing something like this a few years back. In their case, they'd enabled the OmniFocus mail rule - when mail ran rules on their inbox, it would launch OmniFocus in order to run the applescript that's part of the rule.

Barring that, all I can think of would be a cron job, or an alarm set in iCal that launches the app or something like that...

At the very least, you should be able to fix the 'more than one running copy' issue by doing a spotlight search. Sounds like there's more than one version of the app installed on the Mac - moving the older one or ones to the trash should prevent that part of it, at least.