I'd almost think that you might have been bumping an Internet special key on the keyboard if you hadn't said it was happening on an interval. Whether there's anything bad left, who knows, but such things are prone to leave crap behind or broken bits which muck things up, even if the bulk of it is gone. Luckily that's probably the worst of it in your case, because back in like Win98, some spyware would dig itself in to the point that if you removed it impoperly, your internet would completely stop working. The equivalent of taking out a gear between all the rest.
As for the second, that's pretty strange in general. Also makes me think some remnant is left and broken.
Honestly, without sitting down and digging through the machine myself, I don't know what to tell you, if there's even much left you could do. Heck, even if I did dig through it, chances are it's just broken in ways which one would never figure out. I think it's just about time for a reinstall of Windows. After having all that junk on there, it's probably best anyway, since it can be nearly impossible to totally rid yourself of it entirely (as likely seen here) There could even be things left that an antivirus can't detect (like a rootkit).
If you did reinstall, make sure one of the first things you do is get all the security updates from the Windows Update site before doing anything else on the web. Particularly if using IE to browse.
Sorry that I don't have a better solution this time around!