Do you live in your own house or not? You can see what the dog gets up to. You can see what it had got up to when you weren't looking. You punish it for those things and it doesn't do them again. I've trained my cats not to kill birds, not to spray in the house, and not to chew the handles off carrier bags.
They didn't train it well enough. There are plenty of folk who have successfully trained vicious animals.
As long as it's big enough, and it doesn't obstruct the door opening. My front door for example opens against a wall, a large cage would stop it opening fully.