Are antibiotics necessary after wisdom tooth surgery?

Not usually. Most swelling after surgery comes from the trauma to tissue during extraction. This is true of any surgical procedure. Antibiotics are prescribed in some situations, and your surgeon would advise you if needed.
Depends. If the areas are not infected, then usually none are prescribed. Some surgeons, however, like to have you take them just to be covered should anything flare up. Talk with your oral surgeon about your specific needs.