If necessary. If surgery must be done during pregnancy--for example, due to a tonsil abscess--the safest time is during 2nd trimester. But a physician anesthesiologist can give safe anesthesia if necessary at any stage or pregnancy. We use only meds with a long record of safety during pregnancy. If surgery is elective, though, best to wait. Tonsillitis can often be treated with antibiotics.
Possibly, but why? In an emergency, such as appendicitis, general anesthesia and surgery are performed during pregnancy. Both the OB and Anesthesiology professional societies recommend waiting until after pregnancy for elective surgery however. There is increased risk of miscarriage and preterm labor when done during pregnancy. Also, pregnant women have more risk of bleeding due to hormone changes. Better to wait.
No. Most elective surgery should be delayed until after pregnancy. If surgery is urgent it is usually performed during the second trimester (12-24 weeks).
Tonsillectomy. Today, most tonsillectomies are done under general anesthesia. As a rule, surgeries done under general anesthesia are usually avoided altogether during pregnancy unless they are considered life threatening or absolutely necessary. It's a rare case when a tonsillectomy is considered an urgent procedure, although some can be. If surgery does have to be done, we usually pick the 2nd trimester.