What is the ruling on hunting birds with a slingshot or a pellet gun, and is it permissible to eat them, given that they fall dead or mortally wounded?

The ruling on eating game depends on how the animal was killed. If it was killed by its sharp edge, it is permissible. But if it was killed by its broad side or its weight, then it is considered carrion, unless it is caught alive and then slaughtered properly. Hunting for amusement or pleasure is not forbidden if the intention is to derive lawful benefit, such as feeding cats.