Can you catch gray fox in a cage trap? If so, what's the best way to go about it - brush the trap in, bait, etc.?

All depends on the scenario, typically just drop a cage and brush in the bed. Sweet and musky odors with a curiosity food odor works for me.

Yes. They like what you use for Cats

Fox eat and like lots of things. Your abilities in setting your trap site up and recognizing its activity / travel routes will out weigh whatever magic in a bottle of bait and lure you use.

Keeping a clean site set up will help, along with a visual stimulant in the cage. So many things can trigger that curiosity, but one thing is for sure, you'll spend a lot of money trying to look for that magic in a bottle if you aren't up on your trapping abilities.

Magic in a bottle can only do so much.