Does an anesthesiologist have to stay with his patients the whole time while under anesthesia? Thanks

Constant monitoring. I stay with my patients the entire time they are under anesthesia, and care for only one patient at a time. That is the way my group is set up. In other practices, an anesthesiologist may supervise two or more nurse anesthetists or anesthesiologist assistants, so he or she may be monitoring more than one patient at a time. That is called the anesthesia care team model. Either way is safe.
Yes. Anesthesia is continually monitored and so is the patient throughout the entire procedure/surgery.
Not necessarily. Sometimes a nurse trained in anesthesiology will take over after the anesthesiologist md leaves the room. Once the case is not complex and the patient has no medical issues, then a nurse trained in anesthesiology can manage the entire case.
Depends on practice. Either an anesthesiologist will stay with you the entire time, or the anesthesiologist will delegate responsibility to a nurse anesthetists. If this occurs the anesthesiologist will be immediately available for any problems. They will start and finish the anesthetic and monitor it at intervals. Which one that you get depends on location in the us. However, someone will always be there!