I am glad he is eating and drinking, that may be due to his taking steroids, which stimulates them to eat and drink well. Heart disease usually leads to a poor appetite, and a cough related to heart disease will worsen with steroid use, so if it's stayed static and his appetite has always been good then heart disease is unlikely.
Tracheal collapse is something we manage, not cure, but if that is the issue we can do better at controlling his symptoms.
As far as controlling his tracheal collapse, try to keep him as quiet as possible. The more active he is the more he will cough and the more he coughs the more damage he does to his trachea causing inflammation and weakening his tracheal rings further.
Use humidifiers to moisten the air that he breathes as that is soothing to his airways. You can also take him into the bathroom with you when you run a hot shower or bath so he can breathe in the steam.
Keep his teeth and gums very clean with brushing and antibacterial oral rinses (such as CET rinse). The more bacteria in his mouth the more he breathes the bacteria into his airways which leads to more irritation/inflammation and coughing. Sometimes we need to use antibiotics periodically to keep secondary infections under control. I understand that he is taking Doxycyline now and you aren't seeing much improvement, perhaps another antibiotic could be more beneficial for him. Clavamox, Cephalexin or Clindamycin are all other options.
I would add a bronchodilators (like theophylline) and a cough suppressants (like hydrocodone).
Sometimes we need to use oral steroids to reduce tracheal inflammation, which he is taking, but I use them along with a cough suppressant and bronchodilator.
Keep him on the light side, the heavier he is the more pressure on his pharynx, airways and chest and the more he will cough.
I recommend using a harness rather than a collar as that will be less irritating to his trachea, putting less pressure on his trachea.
Over the counter cough suppressants you can try are Robitussin DM or Benylin Expectorant. Use only products with Dextromethorphin or Dextromethorphin and Guaifenesin. Never give those with Tylenol (acetaminophen) or Ibuprofen or a decongestant as those are toxic for dogs. Give 1 teaspoon per 20 pounds of body weight every 6 hours. The side effect most commonly seen is sleepiness. Do NOT use these along with hydrocodone, rather they are a different cough suppressant to try.
A collapsed trachea puts stress and strain on the heart and reduced amounts of oxygen in the blood with a collapsed trachea can lead to heart disease, heart failure, and organ disease. So while the condition itself isn't deadly, unless the airway completely collapses and no air is moving, the stress on his heart can be harmful and lead to deadly heart failure or other organ failure due to low oxygen levels.
Some dogs are affected enough that surgery becomes an option. If he is doing as poorly as you describe that may be the case with him now, but it is worth trying bronchodilators and another antibiotic with a cough suppressant first.
There are two primary surgery options.
One is more invasive and involves putting artificial plastic rings on the outside of the trachea to support it. This requires a longer anesthetic and more tissue trauma when the rings are placed, but the effects from this procedure should last the longest.
The other procedure involves placing a stent inside the trachea to keep it open. This is less invasive and creates less tissue trauma, and it is a very quick procedure and thus anesthetic time is less. The downside is if the stent shifts or breaks it is no longer effective and may cause more inflammation. With the newer stents available this seems less of a concern.
Best of luck with your pup, let me know if you have any further questions.