Hello, my name is***** and I have over 20 years of experience as a veterinarian.
I'm sorry to hear that your fellow has a cough and I understand that you are concerned about him.
I think what you felt in his throat was likely his larynx (voice box) which has small cartilage supports which could feel like a chicken bone. I don't believe he has a bone lodged as if he did he would be gagging constantly and not able to eat and drink.
A cough simply means he is experiencing airway irritation.
A cough can be related to heart disease, including heartworm, lung disease secondary to a viral, bacterial, fungal or parasitic infection or tracheal irritation due to a bacterial, viral or fungal infection. It can also be related to tracheal collapse, seen often in older, overweight, small breed dogs.
In a young dog infectious disease is by far the most common cause. Of those kennel cough or infectious tracheobronchitis is one of the most common causes. If he has been to a kennel, grooming facility, dog park or doggy day care or plays with other dogs kennel cough would be high on my list of possible problems.
Even if he has been vaccinated we cannot vaccinate against all of the agents that cause kennel cough and some dogs will still get sick even with the vaccine. The good news is that dogs that are vaccinated usually don't get as sick as dogs that haven't been vaccinated.
As long as he is still able to eat and drinking well, isn't lethargic, isn't running a fever (rectal temperature greater than 103F), and he doesn't have a green or yellow nasal or eye discharge then you can try taking a watch and see approach and using an over the counter cough suppressant if his cough is interfering with sleep.
Make sure he gets plenty of fluids, lots of rest as the more active he is then more he will cough, and eats well. Add warm water or low salt chicken broth to his food. The better hydrated he is the better he will feel and the thinner his respiratory secretions will be.
You can take him into the bathroom with you when you take a hot bath or shower so he can breathe the steam which is soothing to his airways and thins mucous, making it easier for him to breathe. You can also use a humidifier in the room he sleeps in.
His cough should slowly resolve over the next 2 to 3 weeks. Most of the time these dogs feel pretty well so it's a matter of keeping them quiet and suppressing the cough if we need to.
If the cough is very bad you can use human cough/cold medications with Guaifenesin (an expectorant) and Dextromethorphan (a cough suppressant) only such as Benylin or Robitussin DM can help. Give 1 teaspoon per 20 to 25 pounds every 6 to 8 hours.
If he is running a fever this is almost certainly a contagious infection and if he is running a fever and is lethargic then an examination is definitely warranted as pneumonia may have developed.
If he has not been exposed to any other dogs, has a history of a heart murmur, or hasn't been on heartworm prevention then this may not be related to kennel cough and a veterinary visit to look for primary heart disease or a collapsing trachea is best.
Best of luck with your fellow, please let me know if you have any further questions.