What should we do with this dog?

[Raging Elephant]

I don't see how you adopt a dog without seeing this when you go the pound. Why would you put him down? You'd just be wasting money. You need to be very firm with a dog like this. The people at the shelter might have been more dominant with him. Speak loudly and firmly. Don't need to yell, but you need to remember that pitbulls can be stubborn.

This is all normal for an untrained dog. You'll need to train him. Be firm about it.

Pitbulls have strong instincts to chase small animals, since they're terriers. Try to get him used to the cat, but always keep him on the leash.

When he jumps on the table to grab food or jumps on people, say NO loudly. If he jumps on you, bump him off with your knee. Don't knee him hard, just bump him off.

When he goes in the house, say NO loudly. Don't rub his face in it, don't hit him, because he'll start doing it in spite of you. Make sure to let him out every few hours. If he's stuck inside, he'll go. It would be like me locking you out of the bathroom all day. You'd take a piss in the corner too.

When walking him, if you see him start to pull on the leash, tug back and turn around. If he pulls again, turn around the other way. If he's going off sniffing, walk so the leash goes into a sign. He'll soon learn to watch where you're going and follow you.

Dogs will try to take advantage of you if they think they can get away with it. If he doesn't see you being firm by correcting him (saying no loudly), he'll do what he wants. If he gets the correction, he'll start behaving.

When he whines and barks, leave him. If you go to pet him, he'll learn that if he whines or barks, he'll get what he wants.

Is he neutered? A dog that isn't neutered will run for the door no matter what you do. They're also very stubborn, because their mind is focused on finding a mate. If he is, when he runs toward the door, yell no and stomp your feet. He'll get the message.

A dog is only as good as his owner. If you train him right, he can be a great dog. If you just expect him to come home from the shelter perfect, you're in a lot of trouble. No dog is born knowing not to do any of those things. Leave a dog untrained and he'll bite, jump all over furniture, destroy everything, pee on everything, take food and act like he's the boss. You need to train him.

EDIT: I couldn't help but notice the rainbow in your avatar. You should know what it feels like to be discriminated against. Well, shelter dogs, especially pitbulls or pitbull mixes, are discriminated against too.