Actually, many PAs do surgeries. I'm not saying they do it alone. I'll give you an example; my brother is a plastic surgery PA who would often times do easy surgical procedures such as tummy tucks or implants while leaving the difficult ones for the surgeon. Of course he is "supervised."
You got me with the anesthesia part. Although anesthesia is considered a broad category so technically, they could deliver local anesthesia if they were a certain type of PA, also dependent on the state. So technically yes, they can "deliver" anesthesia.
Here is a link clearly indicating they can administer local anesthesia in Ohio