Background changed on Mountain Lion
If you don't have any Time Machine backups, now would be a good time to make one.

The Desktop Preference in your screen grab is displaying the contents of Macintosh HD -> Library -> Desktop Pictures which appears to be empty.  If it really is empty, you may be able to locate, replace, or download the missing files and that could be a simple fix.  But if there are other issues and it seems that Mac OS X has been damaged, I would think about using Recovery HD to reinstall the OS.  User pictures are located in the /Library/User Pictures folder.  Since Disk Utility has verified and repaired permissions on Macintosh HD, I guess those files are missing too.

Step one: make a Time Machine archive of your Mac Mini on an external drive. Once you are sure it is properly backed up, you could try entering this in Terminal to search the Library folder for the Mountain Lion default Desktop picture:

sudo find /Library -name "Galaxy.jpg"


Or search the entire disk like this (it will take quite some time):

sudo find / -name "Galaxy.jpg"


And just for good measure, run a bit of system maintenance:

sudo periodic daily weekly monthly


If you are still in the same predicament and decide to re-install Mac OS X, reboot holding down the Command and R keys until you see the Apple logo appear.  You should see a list of utilities including Disk Utility (try it one more time) and "Install Mac OS X"  which is a 4GB download, so a good network connection is going to help. After a clean install, you can use Migration Assistant to import Users from the Time Machine archive you made earlier.  Applications are best re-installed from the App Store, original media, or a source you can trust.