I feel nauseous and sick 8 hours after having a mole removal surgery, are my syptoms because of it?

MethodOfAnesthesia?? It is not uncommon to have nausea after twilight or general anesthesia; however, if the mole was removed with you awake, under local anesthesia alone, your symptoms are probably unrelated to the surgery.
Likely due to meds. Your symptoms are probably side effects of medications you received. They could be from anesthetics, even local ones injected into the skin, or from pain medication you may be taking.