The ability can be used on any player's attacking creatures. This includes your own and creatures in an attack you are not involved in (for multiplayer games).

Desert is a land subtype with no special meaning. It doesn't grant the land an intrinsic mana ability. Other cards may care about which lands are Deserts.

The end of combat step happens after combat damage is dealt. You can't use Desert to destroy an attacking creature before it has a chance to deal combat damage.