Your whole body? No. But could you accidentally hit a nerve in your lip and damage it, so that you would have a numb lip the rest of your life? Yes, that is possible.
Sometimes. If you do it by yourself, it can happen. If you pierce it in the wrong spot, you can hit a nerve, resulting in paralyzation. I highly recommend getting it pierced by a purofessional.
Because we have a bottom lip and a top lip so we have 2 lips in total! That means it's a plural! Otherwise if we had a just a top lip it would be a lip!