>>>>Can he give her his leftover pills? Is there anything illegal about this? If she's caught with his prescription, will she be in trouble?
Although possession of prescription drugs is lawful in most cases, possession of such drugs without a prescription or, in some circumstances, even with a prescription, can be unlawful. It is a crime to possess a controlled substance that may only be furnished by prescription without having such a prescription.
Too see how one state, California, regulates this, see Business and Professions Code §4060; People v. Kennedy, 91 Cal.App.4th 288, 291-297, 110 Cal.Rptr.2d 203 (2001) (illegal possession of valium).
>>>>Would the answer change if the prescription was for a narcotic?
The answer would be the same, the offense would likely be more serious.