I figured it out. My home network software is WinGate. When I installed avast!, three applications were automatically added to my Wingate Internet Client to be treated specially: RPCSS.EXE, ASHMAISV.EXE, and ASHPOPWZ.EXE. The special trreatment was set to "local access". I changed all three of them to "mixed access", meaning they are only accessible from my home network, but they can have global access to other machines. Now Outlook Express is able to signon to my account to send and receive mail through avast!.
When I discovered that Outlook Express wasn't even trying to connect through Internet Connection Sharing, as I mentioned in my earlier post, it gave me a clue that my Wingate Internet Client was stopping the process, so I experimented a little. Thanks for helping me think it out.