I saw a post where someone wrote about their puppy/dog coughing and that it had a runny nose. I can't find the post, but I wanted to post that my dog who is 1 year old (Champ) has been coughing now for a few months (but no runny nose). I took him to another vet because the first vet I took him to said it could be allergies, and this vet took xrays of his chest and also his trachia. It was determined that Champ has bronchitis. He's been on meds now for over a week (and he's still coughing but not as much).
Maybe Kennel cough? I got a Shih-Tzu once and I took him to the three times in two weeks and no medication was helping the cough which his nose wasn't runny. So I had to take him back to the owner and get my money back since he wasn't getting any better and was told by the owner that Kennel cough is common in puppies and he would out grow it. Like I was going to believe that breeder. At the time I didn't know about the whole puppy mill thing but I think that is what she was. Now I have a healthy Dachshund that I did a lot of research on the breeder. Good luck and I hope he gets better.