Update:
After a few days without virus protection, I started Avast! by clicking its Desktop icon and changed the one setting that I suspected might have caused the problem:

Program Settings > Common > [ ] (unchecked) Test memory during application startup

I put Avast! through a few of its paces, updating and scanning, then went into Msconfig and re-enabled all Avast! features (three, as I recall) on the Start list. When I restarted the computer, everything came up normally with the Avast! icons back in the system tray.

I held off for another day on entering the changes in my e-mail program that would reroute all incoming and outgoing mail through Avast!. The instructions for making these changes are available through:

Help > Setting up the Mail Program > Manual Setting of Mail Protection

I printed out the section (5 pages) to refer to when I was in my e-mail program actually making the changes. I confirmed with a test mail to myself. The header said (outgoing and incoming):

X-Antivirus-Status: Clean

Some curious adventures during all this:

-- The first two (at least) scans with Avast! found nothing but the latest one, this evening, identified a file in one of my Java folders as infected. Avast! sequestered it in the "chest" where I could look at its details. The problem is classified as "Other:Malware-gen" and the file appears to be timestamped -- tomorrow? I'm a bit surprised that Avast! turned up a problem so quickly, since AVG only spotted one virus in the five years I used it.

-- After the scan, the little icons for my browser and e-mailer disappeared from the taskbar and I had to launch these applications with the Start button. I also noticed that when I'd open a folder such as "My Documents," the little icons for the files and folders weren't showing up. After I rebooted, everything was fine.

-- Right after I'd changed the settings in my e-mail program, I started my browser and it crashed with an "illegal operation" error on the third page. I restarted it and it seems to be behaving normally. Naturally, Avast! gets the blame for everything weird that happens.