First off, I'm slightly worried that you said your puppy is turning 3 months old next week and you got him a month ago, meaning he'd barely have been 8 weeks old when you got him. Puppies shouldn't really be taken away from their mother's until they are 12 weeks old.
Dogs are very limited in the ways they can tell us they are in pain. If removing his paw from the door caused him more pain, then biting you to let you know is going to be his first instinct, that's the only way he can tell you he's hurting.
As for the other biting, well puppies bite, it's perfectly normal behaviour. And the biting/chewing will increase when he starts teething (if he hasn't already). Just make sure you have plenty of chew toys for him.
