Hi there and thanks for your question. While it is possible that there is something irritating your dog's throat, the most common reason for this set of symptoms is that your dog has picked up a respiratory virus such as kennel cough. The good news is that kennel cough is rarely life threatening, but it is very annoying!
Here is a youtube video that demontrates a typical cough that goes with kennel cough although some dogs will even take it further and be retching to the point of vomiting:
If there is something actually stuck in your dog's throat it will cause him severe distress and cause him to not be able to breathe. Obviously if this is the case he needs to be seen immediately. You can do the heimlich maneuver on dogs like in people if he is unable to breathe.
If you think that this could be kennel cough, and it is really bothersome, he should be seen by a vet today. The vet can prescribe medicine to reduce the cough and if they feel like it is necessary then also antibiotics. Many dogs will recover from kennel cough on their own within 2-7 days, but again, if the cough is really bothersome you will likely need some medication prescribed.
I hope everything is ok!
Ah, ok. I would definitely not induce vomiting. If there is a bone lodged in his throat then causing forceful vomiting could cause a tear in his esophagus which would be very bad.
If this is caused by the bone the most likely explanation is not that it is stuck, but that it has caused some damage, or scraping on the way down. (This would be similar to how you feel if you swalled a Dorito and the point scraped you when you swalled, but only worse). If this is the case then your dog may have esophagitis which is inflammation in the esophagus.
It does sound like your guy needs to see a vet. They can prescribe a medication called sucralfate which is a liquid which will help to coat the esophagus and repair any damage and possibly also some pain medication if necessary.
If your vet does suspect an obstruction they will likely do an xray. Hopefully there is not an obstruction because this is a difficult area to get to. If the vet has an endoscope they may do an exam down the throat with that.
I hope he does ok!