Urgh MySQL RPM. I've never managed to figure that one out. I always use the binary tarball.
Your output shows that mysql definitely isn't running.
Change to the root user on your box and then do:
find / -name mysql.server
that should find the file for you.
when you find it change to the directory it is in and run it using:
./mysql.server start
As I've said I personally don't like the MySQL RPM , I'd uninstall it, download the binary tarball and then install it using guide in the README. Works every time for me.
-- Nick Gushlow