I thought that 64 bit Windows could run 32 bit Windows .exe files
(although I don't have a way to check this).
Yes, but there are 32-bit and 64-bit exe wrappers, which I suspect is because a 32-bit exe can't load a 64-bit DLL (and may be vice versa). As I said, I don't know for sure at the moment, but it needs investigating.
Grumble. Next time the label on the can says "worms" I need to leave the can opener alone :-)