Is it bad to punish your dog by putting her in her cage?

generally speaking, YES. Unless it is a room designated for the dog. He needs to feel comfortable and anxiety free when in said room. He need food, water, a place to lay down, and something to chew on. Also dont make this room a form of punishment or every time he goes there he will think he did something wrong and wont like going to the room. Also make sure to walk him whenever possible, and gradually introduce him to the concept of being alone in the room. start with 5 minutes, then 10, 20, 30, an hour, 2 hours, ... until he knows you're going to come back and then he wont feel anxious or the need to act out.