This is unlikely to have anything to do with Flash or Mozilla. (If it helps though, see http://fedoraproject.org/wiki/Flash
for the "official" methods of getting Flash working – you should use the package manager wherever possible rather than copying files manually.)
Also don't try reinstalling software on Linux to fix problems. That's a "Windows" solution, and due to software dependencies will likely break things more on Linux. (Seriously people: stop suggesting this "solution"!)
Do you have the top bar across the top of the screen? If not, make sure GNOME is properly installed (I assume you're using GNOME): 'sudo yum update && sudo yum groupinstall "GNOME Desktop Environment"'. Try again, and if it still doesn't work, check for error messages in the logs. Make sure to choose GNOME in the login screen.
If you have the top bar but no desktop icons, and want them, install and run gnome-tweak-tool, then enable "File Manager"->"Have file manager handle the desktop".