Can upper lip hair removal cause single pus filled bump that starts tingling after being popped, very painful, and doesn't bleed but leak clear fluid?

Yes. It sounds like the procedure induced a "cold sore," or HSV1 eruption and you should be treated with Valtrex (valacyclovir) or Acyclovir asap. You can use Abreva which is over-the-counter. Avoid chocolate and nuts and triy to eat more chicken and yogurt. Don't have any more treatments until it clears up and be sure to pretreat with those same medicines before the next treatment. Also, stop pinching it!
Folliculitus . Sounds like you may have an infection in a hair follicle. Try a dab of antibiotic ointment (like neosporin or bacitracin) and try not to pick it pop it. Get seen if it gets bigger, redder or hotter, or if you start getting more of them. .