It's much harder for dogs who are missing a front leg vs a hind leg, as roughly 60% of their weight is usually borne by the front legs, so it is more to redistribute, and the remaining front leg still bears way more than it's normal "share". Also, due to the hopping nature of their changed gait, that leg tends to take a more forceful impact with each step than they would with 4 legs.
I wouldn't make him walk if he stops and you can't tell for certain it's due to fear, particularly since he's still young and growing. My 3 legger was 50 lbs, and had good muscle tone (she was overweight for a while IMO, but still a normal "pet" weight), but walking 2-3 blocks was enough that she would need a rest before coming home, and usually another break while walking home because she was already fatigued. That was a lot for her, and sometimes she would be sore the next day if she had done other stuff. She was a strong adult dog, so I think it would be too much for a puppy and/or dog with minimal muscle tone. When we first got her, she was emaciated due to uncontrolled diabetes, and it took a while before she really got good muscle tone back, so we just let her hang out around the house and took short walks around the yard until then.
When I took her places that required more walking (she was a therapy dog, so sometimes would go to dog and non-dog events), I taught her to ride in a wagon, and also in a laundry cart. She knew that both meant meeting people without having to do the "work" of walking around, and took to them immediately, but she was a very environmentally sound dog, not afraid of anything, and loved getting out among people. As she got older, and started having a harder time getting around, I started using a harness on her to "help" take some of the weight off her front end while walking, the best one I found for that use was the ruffwear webmaster, because of the extra strap that helped keep it from slipping up when lifted, but any harness might help you to support him and help him out a little.
If you have anywhere that has a dog pool, or are near a body of water that will get warm, swimming is a great exercise for tripod dogs, as it's non-weight bearing. My dog could swim pretty well (and not in a circle!), but hated it, so it wasn't a good option for her. I would introduce him slowly if you do, and use a life jacket because he may tire more quickly, especially right now since he's not in good condition.
For socialization, can you drive somewhere there's some light foot traffic, and just hang out and let him see and meet people as he's comfortable? Maybe a park or pet store during quiet daytime hours. Even if you just take him somewhere and let him look at stuff and reward for brave behavior (curiosity, not shutting down, etc) people don't have to touch or interact with him for it to be a good socialization experience. If he's already making progress, I wouldn't worry too much, it sounds like he's doing fine
