I have a lifetime of experience with dogs and rescues. I volunteered at a no-kill humane society, my husband has managed various kennels and doggie daycares, etc. We know dogs pretty well.
If they don't want to take the dog back, it needs to get into obedience school, stat. Puppies nip, pretty much all of them, until they're taught not to. Big puppies (or dogs) and little ones are hard because the dog can't help that little kids are easily knocked over. Obedience training to get the dog under control can help with that. I have a 65# Viszla/Boxer mix that I found abandoned. He is 10 years old. He will still knock ME over sometimes, if he's really wound up. We warn our friends with children that if they come over, the kids can sit quietly and we will get our dogs to meet them, calmly, and then our dogs (3 of them) go off and do their own thing.
Second, the puppy should be crate-trained. I've never done it with my dogs, but it's not that difficult. Dogs like dens. If a crate is treated like a "home base" instead of a "time out" place, they will come to appreciate it.
Third, as someone mentioned, get that dog exercise. When my big guy was younger, I had to run him. A lot. He doesn't need it as much now because he's old and has arthritis.
Four, there are ways to train undesirable behavior. My dogs hates cats. Therefore, when being walked, he WILL obey me if he sees a cat and freaks out. He knows we're the boss and he will sit on his butt and calm down before he's allowed to do anything else. When he behaves, he gets a treat. If he continues to spazz out, he has to sit and watch me...which takes all of his energy. But he is leaps better than he used to be. I'd never let him around cats...but my 5# chihuahua is the boss of him.
Rescues are evaluating dogs in a shelter, which is a far cry from a home environment. Those who rely on fosters have a better sense of the dogs' behaviors, because they're in "real" environments. A shelter is a tough place for sensory animals like dogs. It's noisy night and day, dogs are stressed, others feed off that stress.
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