Re: chronic dry cough
The most common causes of chronic dry coughs are allergies, asthma, PND and GERD. Since you've ruled out the allergies and asthma, looking to the GERD sounds like a sound next step.
I had a chronic cough as well. I worked with my Pulmonologist, Gastroenterologist and ENT to get it under control. (The ENT helped with the allergy cause.)
I'd take a look at your son's diet to see if you find triggers. Things to avoid include: greasy, fatty, or fried foods, spicy, tomato or citrus based-items, anything acidic, caffeine, chocolate, carbonation. I also found avoiding dairy helped.
Sleeping on an incline might also help. 4-6 inch rise in the headboard may help. I use those plastic bed risers. My docs recommended stacking a couple bricks.
My research shows accupuncture sometimes helps with the digestive issues. I haven't tried that though. I do go to a chiropractor, but I can't say I can link his treatments to any improvement in the reflux.