What type of screws and drill bits should I use to mount a mantel over faux stone?
A regular high-speed steel bit will probably cut through the soft faux stone just fine, though it'll dull the bit fairly rapidly. If you have just a few holes to drill I'd go that route. 

Otherwise, a masonry bit will also work until you reach the wood. At that point you'd switch to a high-speed steel bit and try not to bung it up on the way through the stone--maybe insert the bit fully before you start spinning it. 

The holes in the stone should be the size of the screw thread diameter to prevent undue damage to the face of the stone. The pilot holes in the wood should be slightly smaller than the screw shank.

You want screws that are intended for structural use. Black oxide drywall screws are too brittle. Use gold construction screws or something specifically described as "structural" which are of suitable size and length. Unless your mantel is obscenely heavy, just about any screw #8 and up should do fine.