Mint will provide Flash through the package manager, or software centre or whatever it's called. Or, as mentioned, you can apt-get it. The Firefox warning normally means that Flash is not up to date under Debian I sometimes have to run the following (as root) to get it up to date:
Code:
update-flashplugin-nonfree --status
Thereafter the message in Firefox disappears.
I don't think you should have to do that on Mint so you should just have to ensure that you install updates regularly.
Whichever, either way you should not have to download anything manually from Adobe. If you've carried out a system update recently and you're still seeing issues let me know and I'll try to install Mint in a VM and check things out for you.
Regarding Linux support for Flash -- it's supported for a while yet though only with security and bug fixes and Firefox does sometimes give an error that Flash is out of date before Adobe get around to fixing it. However, as things stand, you shouldn't be seeing problems with Flash in Firefox if you keep your Flash up to date.