I think you're right that it was the spyware programs that told me to disable the system restore.
After you clean out your spyware you should re-enable system restore in my opinion. It has been a God's-send for me a couple of times. I recently downloaded Microsoft Windows Media Viewer version 10 and it wouldn't work and wouldn't uninstall. But System restore took me back to a usable version 9.
Since the desktop works from the same wall connector, your laptop is clearly the culprit here. And if the modem is checking out on the troubleshooter, then you might look at the tiny connector wires at the outlet where the phone line plugs in as the problem. Any dirt or lint? Bent?