Have a look in the data directory (probably /var/lib/mysql/data) for the .err file. This should tell you what's going wrong. At a guess I'd say that MySQL doesn't have permissions to write to /var/lib/mysql. You probably don't want to mess with the permissions on the mysql directory (bad idea!), but try changing your /etc/my.cnf (or /var/lib/mysql/data/my.cnf) file to put the sock and pid in something like /tmp.
Have a look in the data directory (probably /var/lib/mysql/data) for the .err file.
By default, this is not in /mysql/data, but in /var/lib/mysql, nothing more. However this is going to be the best place you'll find answers to your questions. Look for hostname.err open it up and see what the problem is.
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