Do they use an anestheic when circumsizing a 2 month old baby?

Some do some don't: Some physicians use topical Emla (lidocaine and prilocaine) or lmx topical anesthetic cream, applied about 20 minutes ahead of time. Some cover with tegaderm or the end of a finger of a surgical glove to keep cream in place. Some inject a little Lidocaine as an anesthetic block. We have found topical cream to be very effective plus "sweet-ease" sugar water for baby to suck on. Sucking releases endorphins and reduces pain.

Yes: The surgeon may opt for a general anesthetic. Some will use local anesthesia with sedation.