This has nothing to do with OE. I am having the same exact problem with Thunderbird since applying the 4.8.1201 Program update. Once, I was able to stop and restart the Internet Mail provider and get it to work. Today I have been unable to restart the provider. I, too, also had the Internet Mail icon in the taskbar stay even after I had canceled retrieving mail and even exited Thunderbird. I tried to stop all of the providers (Stop On-Access Protection), and instead of all of them stopping it attempted to stop Internet Mail once, then gave up (all other providers were still running). Trying again to stop all On-Access Protection finally did result in all of them stopping (after an abnormal delay, at which time the Internet Mail icon finally went away), but now the Internet Mail Provider will not restart. Attempting Termination seems to work, but upon starting it never gets past a status of "The provider is waiting for a subsystem to start." Mail retrieval times out and fails.
I would be surprised if I had a spambot on my system, since I scan regularly (weekly on average) with Ad Aware, Spybot, and even Windows Defender (which I mainly use for real-time protection since I have had problems with Spybot's Tea Timer). I also have not noticed any unusual system behavior or excessive utilization of my Internet connection.
I never had a problem before the 4.8.1201 update, and now this has happened to me twice. I have moved the sensitivity to High, just to be sure, and will monitor outbound connections on my system. But this smells like a fairly nasty little bug in Avast to me.