Tried your tip, but I'm not seeing any activity on ports below 1000 with the Avast mail plugins disabled. And most of the eD2k activity is on the ports assigned to them in the program itself.
When I enabled the Internet Mail plugin I see it using the pop3 protocol, but it doesn't trigger any connection problems.
When I enabled the Outlook/Exchange plugin I see activity from Ashmail on 2985 that goes away if I respond "no" to the timeout message. So I believe that's the root of my issue. But I don't see why Avast would be using 2985, as its not an open port in my firewall, and not a port I would anticipate needing to open.
Opening that port seems to have resolved the problem. [edit] That only helped temporarily, the port it has conflicts on seems to change.
Also, I'm seeing a connection from Ashmail to a dial-up user on shawcable.net. Any idea why this would occur?
ashMaiSv.exe:3376 TCP phydeux:3194 s01060004e2504e41.du.shawcable.net:pop3 ESTABLISHED