I have no problem in Ubuntu 64-bit with Flash in Firefox or Chromium (or anything else really these days).
The trick with Flash is not to use the default (dreadful) 32-bit-plugin-with-wrapper solution, but to download the 64-bit Flash 10 ‘alpha’. It seems to have been called alpha to avoid offending the other platforms where there is still no 64-bit build at all, but it is in my experience just as stable a Flash as on Win32 (...still not perfect, obviously...) and way ahead of the wrapper fudge which Ubuntu are still using out of fear of the word ‘alpha’.
However you're still out of luck with AIR. There is no 64-bit AIR (come on you commercial vendors, I know 64-bit applications are still not widespread on Windows, but is it really that difficult to do a build?). Though it is supposed to work it looks like a huge mess and I'm not greatly surprised if it has the same stability problems as the 32-bit Flash Player hack. If you absolutely require AIR, you may be better of with a 32-bit OS for now.