John, that was my thought - that it might be a program checking for updates.

If so, one of those things you bought from new stores is likely the culprit if they were new software or upgrades. You ought to be able to open Zone Alarm and see which program is trying to access the internet.

My anti-virus is set to download updates daily, whenever I'm on line. The first time it did this, I saw something fairly large downloading and mildly freaked...until I checked Zona Alarm, saw that it was the AV's Instant Update, and remembered I'd told it to do that. That's the "Hey, everything's fiiiine" view.

Now, for some paranoia:
"...the connection it wants me to make uses my regular earthlink number, but does not make a useful connection - can't use mail or netscape from there." gives me the creeps. There are things out there which will link you to THEIR server through your connection. I don't know what that looks like when it happens, though.

If your machine was ever infected with a virus, you would have had to not only delete the virus, but also undo the changes it had made. It doesn't sound like you ever found any viruses on your system, BUT I wonder if the same thing's true of spyware. (That it makes changes to your computer registry which you have to undo.) You also should check for updates of the Anti-Spy stuff regularly.