Eating vomit is pretty normal. They see it as food. It's not nasty to them. Just to us.
Has he been fully vetted?
At almost all animal shelters and SPCA's, there is one commonality when you take the dog home. No matter how clean the shelter is or how well maintained, most of the dogs go home with kennel cough. It's a hacking sort of cough/nasal/eye discharge/sneeze that can go on for weeks. It may also present as a gagging sound.
It is contagious to your other pets.Some dogs get over it on their own while others need antibiotics. If it's kennel cough, the vet can provoke the cough and tell you for sure with a little touch to the throat.
Keep in mind that kennel cough is a virus so antibiotics won't help it. HOWEVER, there are often secondary conditions that come with it and those MIGHT need antibiotics. Pay close attention to the discharge from the eyes and nose. If it's green/yellow, your pet needs antibiotics. If not, and he's eating well and seems to be of a normal activity level, you can wait and let it run its course.I hope this helps!
It's hard to say. Some dogs can pass small objects and others can't. I would add some fiber to his diet like rice or bread. You can even use canned pumpkin, which is full of fiber. This will move them through.
If the vomiting gets worse or the cough does not improve, don't hesitate to see your vet.