I THINK we're fixed. It only took the entire day (excluding church service).
I had to download a freestanding virus scanner/cleaner on another computer, onto a thumb drive. Ran that -- it found 1524 infections, and (of course) my wife has no idea where they came from or how they got there. After that was supposedly cleaned up was when the executables wouldn't run, so I went back to the oldest restore point available (March 22 -- which wasn't old enough) and restored. After that, Malwarebytes would run, and it found and removed 177 more infections.
But then Avast! (my anti-virus software) wouldn't run. So I uninstalled that, downloaded the newest version, and reinstalled. Things now appear to be running about normally (which, given that it's a Vista computer, means about as fast as blackstrap molasses on a cold morning). But we're in business .. I think ... and hope.
It was disappointing that none of the "solutions" I found through Google worked. Depending on which one I read, there were differing lists of what to delete -- or whether to delete registry entries or just change values. All of which was meaningless, because the instructions for how to get there and do that ... didn't get me there. Probably just as well.
I'm sort of thinking I might take the $100+ I didn't spend on taking it to a professional to buy a 3-copy family upgrade to Windows 7 Home Premium. It has to be better than Vista (doesn't it?).