CAN I ASK MY VET TO CUT OFF MY DOG'S JOWLS?

No, you can't. That would be the most unnecessary of unnecessary surgeries. And your dog would just drool right out of her mouth anyway. Pets are like kids -- you have a commitment to them even if they do things that are gross sometimes. Keep a little stack of handtowels or a roll of paper towels near where you sit when your dog puts her head in your lap, and learn to live with the drool.