^Yes.
In regards to DURING the procedure, that will depend. If your teeth came in normally it usually won't require being put under. So more likely just novicaine, maybe nitrous oxide as well to relax you. But if your teeth are impacted (sideways), then they have to do a little more surgery and will put you under, usually with a combination of fentanyl/midazolam. This will keep you out, and take care of the pain during the procedure as well.
Afterwards I would expect maybe a week's worth of vicoden/percocet if you are lucky. Possibly codeine. since there will be swelling afterwards, taking ibuprofen is actually quite effective as it is an anti-inflammatory.
Most people have them out when they are younger, but it isn't abnormal to have them out during your 20's. I had them out when I was 21, and had two impacted teeth that required being put under. It wasn't very bad at all. Sore for a few days, but nothing heroin couldn't take care of, lol.