Hi, I have a bijonpoo and he is very territorial, he has bitten my mom the first day we got him (probably because he didn't know her and she was trying to take something out of his mouth), then he bit me because he was humping something when I tried to hug him.. and now he bit my nannys daughter.

My mom is thinking of giving him away, but I want to keep him... is it normal for a dog to be this territorial, or do we have to give him away?

The problem here is that this dog is the boss, not the other way around. This sounds like dominance related issues to me. This is a problem that can be overcome with consistent training.

I would recommend Caesar Milan's book, Be The Pack Leader.

When training your dog you need to use strong, firm commands, but without harsh physical punishment. Hitting the dog will cause the animal to feel scared or threatened and may have dangerous reactions to these feelings.

A clicker trainer might also help this process.

Also is the dog neutered? If not then that may also be part of the problem and you should put heavy thought into having this done.