I haven't tried it yet with the GN10.1, but I use Dolphin on my Galaxy Nexus, and the option to use Flash actually pops up saying it is not supported on the OS Version so they do not allow it, to prevent any compatibility issues. Maybe it's just Dolphin and Chrome that don't want to play nicely with Flash. I hate the stock browser, and haven't used Firefox since the terribly bad earlier versions.
FWIW- Before I updated the GN10.1 to JB, I made sure to go and download Flash and Air from the Play Store, then ran the update. Following that, Flash still worked in Dolphin (I think it was still version 9.?) When I updated Dolphin is when anything Flash based stopped working...
I too have tried to sideload Flash using the methods above, to no avail. I will give Firefox a try though...