Control 12V LED circuit and 120V power supply from one switch
LED's are run by drivers (ballast by another name), but they are electronic. Electronic drivers and ballasts are not damaged even if they are running in no load. You are referring to the old axiom about magnetic ballasts which were a type of autotransformer. And an electronic ballast can sense load and turn off unlike a magnetic.

Part 2, you can have 12V and 120V or any other voltage combination in an enclosure so long as the insulation has the same voltage rating, otherwise there must be a partition. I would assume by looking at the driver that it was made to go into an enclosure with no partition. 

Now to your question. I would assume that if the driver was able to carry two separate switched loads, it would tell you it could and provide a drawing on how to do it. Try going to their website and see if they have a specification and/or installation paperwork. Even though the NEC wiring methods usually apply to voltages over 50V. I would not switch the return wire. The first reason is because if another electrician was to work on this he would assume that it was connected under standard methods causing confusion. The other reason would be that everything up to the switch would be hot even though it appears to be off leaving a chance for a shock or burn hazard.

Stay safe.