Have you taken a look at the send and receive settings under the tab for advanced options? I have mine set to automatically poll all accounts for new mail every five minutes.

Yes, that's exactly what I have mine set to.

Another clue: If I leave a folder open in outlook THAT folder will sync automatically, but the others wont. I can tell the others aren't syncing because I don't see a new message count next to them. Once I click on a folder, then that folder will sync. However in no case do the rules ever run automatically.

I haven't any idea what might be causing that. If you haven't already done it, you might want to try either the off-line quick repair or the on-line Office repair option through the control panel. On-line option claims it fixes all problems. Couldn't hurt to try.