I think the problem with playing flash results from using swfdec as flash player and not the original one from Adobe. Adobe doesn't licence its flash player under a GNU-like licence, so the GNU project has developed a replacement (which is incredibly slow, though). You should be able to install the Adobe player with some trick through your distro's package management (a google search for "fedora install flash player"
should get you going).
I can't think of what the problem with your ATI drivers could be, maybe another post "ATI drivers not working" or something would be a start there. Include a copy of your /etc/X11/xorg.conf file, and other stuff people will perhaps be asking for