Wow, tough situation.
I guess, I would want to know if the neighbor went to the hospital and had stitches or a drain put in. This is something that could indicate whether this was a nip or a bite. As you know GSDs have gigantic teeth and if they are intent on doing injury, they will cause bruising, bleeding, and often the need for stitches or drains. This medical care should be paid by your family, and apologies are in order.
It would be good to know what was happening at the time. For instance, if the lady was walking from the car to the house, and the dog ran up and nailed her, or was the lady having her morning run and the dog chased and nipped. This is not to make the neighbor liable for the bite, but to give accurate information to the behaviorist-trainer that you employ to evaluate and or train your dog with you. The behavior can mean different things. GSDs do not generally roam around looking for someone to bite. A running person might be too much for a young dog to resist, and it if was a nip it might have been communication -- "And stay out!" or "play with me!" Could be. Or it could be a dog of lesser character who will chicken-bite. They wait until whatever it is they are fearful of turns their back and then they rush up and give a nip or a bite.
From the neighbor's viewpoint, a bite is a bite. From the owners, it is more complicated. You have to decide whether the dog made a poor choice, and improved confidence, increased training, a change in leadership style, and better management will mean that you can safely keep your dog, or whether the dog is aggressive to the point that he is a liability to children and adults and needs to be put down so that he doesn't cause a more serious injury or death of someone.
I don't think you are there yet. That kind of decision should be made by someone who is well-versed in the breed and is evaluating the dog in front of them. The dog is a young dog, just getting through the teenager-stage. It could just be a young adult who is full of himself, with sketchy leadership, and poor management.
Management -- first you got to keep others safe as well as your dog. I agree that 24 hour muzzling is not an option.
Leadership -- NILIF -- Nothing in Life is Free is one leadership style. There are others. Consistency and confidence are key here. The dog needs to trust you to protect him and look to you for how he should act.
Training -- Every dog needs training, but GSDs more so than some. They are intelligent, energetic dogs that are not necessarily an indicator of docile obedience. The bond between owner and dog is built and solidified through training. Without it, the dog has little structure, and does not know how to act in all situations.
Exercise -- A young dog who is bored and left to his own devices is trouble. This dog should not have access to the gate open or not on his own. He does not need a ton of freedom, he needs to be taken out on lead and walked, maybe run, and regularly. The gates at home should be secondary defenses. But the dog should not be left to exercise himself -- that is your job.
Socialization -- This is last because a dog should be well-managed, under a solid leadership program, trained or in-training with confidence in his handler and well exercised before we expose them to people he doesn't know, and situations that may make him uneasy. Puppies are usually much better equipped to experience a variety of things, and this should be done during the early puppy stages to take advantage of that and to set some of that learning. If that was not done doing it as an adult should be done only after great strides have been made in the rest of these.
I hope you can find a good trainer-behaviorist who is willing to evaluate your dog and work with you and him.
Heidi Ho, Odie
Joy-Joy, Bear Cub, Hepsi-Pepsi
Cujo2, Karma Chameleon
Ramona the Pest, Kojak -- who loves you baby?
Tiny Tinnie, Susie's Uzzi, Kaiah -- The Baby Monster.