Thanks for the additional information. It is helpful.
I'm going to go over the various causes of coughs and let you know which are more likely than the other. Dogs can develop upper respiratory illnesses such as kennel cough and canine influenza as well as bronchitis and inhaled allergies. Just like humans dogs can get bronchitis. You can read about bronchitis here: With bronchitis there is often a productive cough (spits up mucous).
Dogs can also be allergic to inhaled things like pollens, air fresheners, cleaning supplies, dust, etc. This will often cause sneezing and nasal congestion even some eye issues. If this is the case, Benadryl can be given to your dog, the dose is up to 2mg per pound every 8 hours. If you try benadryl and it reduces the cough, I'd look for new cleaning supplies etc that you have only recently started using. You can read about these here.
Kennel cough is normally contracted when a dog has been boarded or kenneled or around a large number of dogs such as at a dog show, dog park or pet store. In most cases an adult dog will have it for about 3 weeks before recovering without treatment. Often a vet will supply antibiotics if it is a pup that has contracted kennel cough. Kennel cough generally lasts about 3 weeks in an adult healthy dog. Now if your dog got a bordetella vaccination, then he may have developed a mild case of kennel cough. This does sometimes happen and the vaccination just protects against the most common causes of kennel cough and not all causes of kennel cough. Here is a website with more information on kennel cough.
Canine influenza is now becoming more prevalent and like it sounds it is a canine flu. This is contagious so the trip to the vet may have exposed your boy to this. Here is an excellent site on it.
You will want to monitor your dogs condition looking for colored discharge from the nose or eyes, a productive cough (coughs stuff up), stops eating or lethargy. These are signs of a possible bacterial infection as well and may require antibiotics. If your dog appears to be having a difficult time breathing, you will need to see your Vet as some dogs dog get really sick with canine influenza and need support to recover.
To help your dog breathe easier you can run a NON-medicated humidifier in the room your dog is in, or sit in a steamed up bathroom with your dog to help keep the mucous moving. Robitussin DM at a 1/4 teaspoon per 5 pounds can be used to control the cough. It should only contain dextromethorphan and Guaifenisen. and should NOT contain pseudoephedrine. If your dog has not had a vet visit in a while or a heart condition is suspected you should not give Robitussin until it is ruled out as it can mask heart issues. Dosages can be obtained at this website.
There are other reasons for coughs such as heart problems which should be investigated if your dog is not on heartworm preventative or is an older dog. The following site goes over canine coughs.
You say it is not a reverse sneeze but could be a collapsing trachea. Here is a site on collapsing tracheas. The dog does cough with this.
He does sound like a typical mini-pin. Such an alert breed and usually pretty fun. In this particular case, I'd try the benadryl and if the results are positive, use it for a week or two while you determine what might be the cause of the allergy or it is a pollen, it eases up. If the benadryl doesn't work then you might try the robitussin as long as you are sure he does not have an undiagnosed heart issue. The medication masks the heart condition so interferes with determining cause if you take him into the vet. So if you plan on taking him in wait until the vet examines him before using the cough medication. If it is a mild case of kennel cough, the robitussin can help control the cough until he recovers.
If neither seems to help you or your dog develops additional symptoms, you would need to have him seen again.
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