I branched this into a new discussion because I wanted to dig into it more deeply with you.
What you're describing is definitely not intended behavior, but I can't quite get my head around what you're actually describing. Can you elaborate on what you mean about "it wants to download and reinstall every time I restart Firefox"? Are you getting an update prompt when you restart? Can you provide a screenshot? That would be super helpful.
It might be worthwhile to just uninstall and reinstall to see if we can get you out of this busted state.
1.) Reboot your computer. For real. Reboot.
3.) Reboot again.
4.) Download and install Flash Player from http://get.adobe.com/flashplayer
If you're still stuck, please let me know. I'd still like to get a picture of the dialog that you're seeing. That's super weird.