Desert Combat installs itself in the Mods directory in the Battlefield install dir.
To run it, you can use the supplied shortcut (which should appear in your "start" menu.)
If you started the regular Battlefield, you can change into DC (or any other mod or expansion) by clicking the "customise game" button in the top-right corner of the main screen. It will then load the appropiate mod.
As for singleplayer: in DC not all maps are supported in single player mode. Try a multiplayer game with them to see if it is really an error, or go to the Desert Combat website:
LINK
Edit: as Mephex said, make sure you've got the latest version of DC and Battlefield itself.