Can getting baby teeth out at the dentist hurt?

Should not. Although there is a small amount of discomfort inherent in an injection to numb the area, once it is numb, removal of baby teeth should be painless. However, there is no local anesthetic that can remove the sensation of pressure, which you will feel.
It shouldn't. If numb only pressure is felt. We don't mention shots or pulling teeth to the child (at home or in the office) and most often the extraction can be done without the child knowing it. We don't fib to the child--we just explain what we are doing without using frightening words. We are "fixing the tooth" or "squirting sleepy juice" in the mouth. If parents are over anxious the child can tell.
Shouldn't. Thats why dentists are so good at providing anesthesia. Once the tooth is numbed up, there should only be a very tolerable feeling of pressure as the tooth is removed. Dentist will then give you post-extraction instructions to make you comfortable.