Okay, it took me a while to get to try this, and there was a lot to try. Thanks again for your help.
Removing the Plugin Path line from operaprefs.ini had no effect (and there is no handlers.ini on the entire machine).
I fiddled with the .DLL versions inside plugin-ignore.ini and on the hard drive; Opera is not blocking all of them indiscriminately.
Everything is 32-bit: the CPU, the OS, and Opera itself.
I fiddled with the .DLL versions inside the system folder where the newer Flash DLL is, it will only pick up the newer .DLL. If I tell it to ignore the newer .DLL, then it uses nothing.
I tried renaming the operaprefs.ini as "handlers.ini"; no luck there either.
The only thing that comes close: I renamed the v10 .DLL to the one that Opera seems to seek out, and that tricks Opera into using the v10 .DLL. The only problem with that approach is that Firefox is trying to use that same .DLL (registry?). Again, I was hoping to get one browser to use one Flash version and the other browser to use a different Flash version, since some Flash versions play videos better than others, and vice versa.
I really appreciate your help, but I am thinking of throwing in the towel and trying a different browser. I'm usually very persistent, but I know I am fighting a losing battle as I try to keep this single-core 3 GHz machine going for my folks. You would think 3 billion operations per second would be enough to watch some web video, but not if Adobe's bloated software has anything to do with it, apparently (BTW, I am a programmer). I have been an Opera fan from so long ago, that I actually PAID ($40?) to have its built-in ads removed (version 4 or 5?), so I was interested in hanging on but there's only so much time an adult can dedicate to "the trivial". Thank you all for your posts, especially @burnout426.