Hi! I really hope your baby feels better soon!
First off with the vaccines I would go with that he hasn't had any. Even if a breeder does their own shots there are stickers that go with those shots that you would put on the paper to "prove" that they were given. I am amazed a vet would say ok with that paperwork to ship him.
Loosing hair is pretty normal in a pup that has been through so much. The body protects important organs first and hair is not that important and if in bad cases the hair will stop growing and will fall out. Don't worry - it will grow back just fine.
I see you already checked for mange so I would think it is just a case of bad nutrition. Does your vet have fortiflora to add into his food? It's not cheap, but will help by adding probiotics (good bacteria), vitamens and more. They usualy really like it too.
Depending on how bad of a case of cocci do you think his blood sugar dropped and caused a seizure? This can also happen. Or was this gone by the time this happened?
I would really want to look more into that coughing. Just doesnt sound right to me. Kennel cough causes runny nose too? You can bring him in the bathroom with you when you take a shower to breath in the steam and that might help or the those things you use with kids when they are congested (sorry cant think of the name). I hope some of this might help!
Hugs to your Wiley!