I am sorry to hear that your dog died. Please accept my condolences. Unfortunately, I will likely be unable to shed much light on why, having no direct interaction with him.
Nor can I really comment on your veterinarians competency. Basic competency is assumed when the government grants the license to practice. Of course some may pick up on subtle signs quicker than others and of course we are all human and make mistakes.
It is not uncommon for puppies to contract infectious diseases, many of which we vaccinate them for and indeed we vaccinate their mother's for, before they have given birth, so that maternal immunity passes with the first milk, long before vaccination is possible.
Kennel cough vaccine would not lead to death under any circumstances I can imagine.
The vomiting that began on 12/22 which included mucous similar to that in the stool may well have been due to an infectious disease. Puppies, especially small ones are at risk from such viruses due in large part to the dehydration and low blood sugar that may occur with vomiting and diarrhea. Persistent vomiting and diarrhea, like occurred here, often warrants hospitalization for intravenous fluids to counter those consequences. By the time the vomiting reoccurred on 12/24 it is reasonable to assume dehydration existed and perhaps low blood sugar.
Finally when a dog vomits, especially a puppy it is possible for them to aspirate, suck fluid or food, into the airway and lungs. This is called aspiration and may cause death quickly. It is possible this ocurred with your puppy, secondary to the vomiting. You mention a respiratory problem, but describe no symptoms. If you mean the kennel cough vaccine, it is very unlikely that played any role here.
Either the pup died due to a viral disease or other insult causing vomiting, which lead to dehydration and low blood sugar or perhaps aspiration occurred.
Again, I am very srry for your loss.
Arnold L. Goldman DVM, MS
MedHelp & Pet Docs On Call