Does the anesthesia affect on fetus? My wife did an operation for Bartholin's abscess.

Variable. This depends on the age of the fetus. Talk to your anesthesiologist to discuss your wife's case.
Doubtful. Depends on the type. Having said that, both local and general anesthesia are very safe, and unlikely to cause problems with the baby. Just like everything however, there are no absolutes. See your OB doc for more info.
Anesthesia . Yes...It depends what was used however. She really could have had the whole thing done under local anesthesia probably which has minimal risk.