What are the laws behind hunting? Does it make a difference if the hunting (let’s say for deer) is for population control purposes?
Hunting deer is not allowed.
As a general rule mere sport hunting is not allowed. Since deer are not a danger to humans as a general rule, it would be difficult to argue that hunting would be permitted to regulate population control.
See the Noda Be’Yehuda (Tanina Yoreh Deah 10) that prohibits hunting for various reasons, of them: 1) it is dangerous, 2) it is causing needless pain to animals, and 3) it is wasting, Bal Tashchis.
Even if not all three reasons for sure apply, at least one of the reasons should apply in any given hunting case.