I've moved into a new location and I have 12" studs rather than the 16" studs my TV mount was designed for. The TV mount comes with lag bolts (M8x55). Getting the lag bolts and pilot holes drilled don't seem to be a problem; the mount is around 25 lbs and my TV is around 35 lbs. I know that in a broader situation (e.g., 24" studs) I'd want a plywood backing behind the mount. Am I correct that I shouldn't need one in this case with the studs being closer together as long as the lag bolts are correctly placed in the center of the studs?
You haven't told us what mount you're referring to, but most aren't designed for specific stud centers. Instead, they have a range of available mounting area (and many will accommodate 24" centers as well). It's very likely that your mount will work just fine on 12" centers.
You should be able to put a tape measure on the mount and verify. Can you position two bolts 12" apart? If so, the only question is whether you'll be able to mount the TV exactly where you want it, or if you'll have to shift to the side a bit.
Yes it will work on anything less however you might want to move it one and a half inch right or left. With commercial toggles it doesn't matter. They hold up to 280lbs on drywall.
Also remember you will usually have 3-4" of play. If articulated then you need to be center unless your willing to a lot more for a adjustable wall mount plate.