“Does God approve of the death penalty?”
The Law of Moses governed Israelite society in one form or another from around 1450 BC through AD70, when Jerusalem was destroyed and the Jewish people forcibly dispersed throughout the Roman Empire. In that law we find a plethora of offenses that call for the death penalty, among them murder, kidnapping, child sacrifice, rape, witchcraft, blasphemy, false prophecy, profaning the Sabbath, violence against father or mother, adultery, bestiality, homosexuality and perjury.
Did God approve of putting people to death for these things? Of course he did. The Law of Moses was the expression of God’s will for Israelite society. It came directly from him.