Will I get pain medicine after tooth extraction?

Depends. It depends on the difficulty of the extraction. There are times that advil or tylenol (acetaminophen) is sufficient. If your dentist feels you will need something stronger then he will give you a prescription.
You Should. It depends on severity of the extraction but pain control is most always provided after the procedure. It could be a number of different things, just ask your provider aobut it.
Yes plus ice, etc. Any anti-inflammatory medication can help. Aleve, (naproxen) motrin, advil and even a holistic alternative arnica. Ice to minimize swelling can also be helpful... 15 minutes on, 15 minutes off. Loritab or something similar is usually prescribed with tooth extractions, for patients who have no allergy to codeine. A very gentle salt water rinse after 24 hours can be soothing and healing...