Welcome and thanks for adopting a special guy. I love the shoe.

I adopted my girl when she was just 2 months. She is blind, brain damaged and had a stump for her right front leg. If it is is at all possible I really recommend you find a veterinary physical therapist (that might not be the right title, my brain is drawing a blank). One of the things she can do is give you advice on your dog's gait. You want to be careful about encouraging Auggie to use the stump if it's throwing off his natural gate. That could lead to other issues creeping in. Dogs do really well on 3 legs, so he  may actually do better without. Definitely wait for the prosthesis 

As far as exercises have him walk over stuff and around stuff. Wobble boards are great and he is young enough you can probably train him to like it. I live in a woodsy area and the river beds around here are rocky so after a few visits with the physical therapist I figured out that Cora was getting the same experiences as the obstacle coarse just navigating our area. Anything that helps him with balance is going to be great. Also, you might to learn how to massage him and stretch him. That's where the PT, Cora saw really taught me a lot. 

As he is a puppy, it will be important to slow him down and create rest breaks. Be aware of what he is doing. He is going to have all the energy of a quad dog and he'll amaze you with what he can do but he is putting extra work on his other joints. And you are spot on about protecting that stump. Cora ended up having hers amputated because she injured it. It turns out it was causing her pain anyway, so it turned out to be a good thing. 

That was a lot of info I gave you. Don't worry, dogs are amazing and he is going to amaze you with his adaptability