I have a morbid fear about pulling wisdom teeth because of some horror stories I have heard before -- Someone underwent a very extensive surgery to get the deeply buried wisdom tooth out and ended up with damage to her facial nerves. So she could not feel half of her face, permanently!
General anesthesia just knocks you totally unconscious. Some people require a smaller dose than others, but it doesn't have anything to do with physical tolerance upon repeated use (ie, frequent use of anethesia... Wait, nobody undergoes anesthesia "frequently"

), but has more to do with individual physiology.
Pain killers are a separate thing. You may or may not need a pain killer to go with the general anesthesia. I am no expert so I can't say for sure.
Local anesthesia, like lidocaine that dentists often use to numb a small area in your mouth, blocks all sensation of pain, but you still have the command of your jaw and tongue because the effects are local. This is not the same thing as ordinary 止痛药. Ordinary oral 止痛药, including Tylenol, ibuprofen, morphin, opium, codeine, is always "general", which means that any pain signal from any part of your body will be blocked in the brain. Usually it's not necessary or efficient to use 止痛药 during the procedure if it's short and local, but after you go home, you can't keep injecting the gum with lidocaine yourself, so you can take 止痛药 for the prolonged pain and discomfort.
Whether the effective dose of anesthesia is in anyway associated with the tolerance of pain and pain killers -- I don't know. Possibly but I don't know. Physiologically they are two separate systems, somewhat.
Also, how quickly lidocaine wears off is definitely determined by genetics. It has to do with the amount of a kind of enzyme you have, which "digests" lidocaine or other types of anesthetic agents.
OK, enough of the pharmacology crap.
A funny factoid about pain tolerance is that women are more sensitive to pain during the menstral period. I don't know why, but that's what the data says. Women in general have a lower threshold for pain.
A person's pain threshold is probably also genetic. So the logic goes this way: You have a high threshold for pain, so you use less pain killer in general and require less pain killer in surgery as well.
When people say to patients with chronic, undiagnosed pain, "It's all in your head," they are not wrong. All pain is in one's head.