I'm also having a similar problem. However I don't cough after the deep inhale, just feels funny on that first deep inhale. It also doesn't happen when I'm lying on my side, only on my back. I had all sorts of tests,(xray,spirometry,blood and urine, thyroid) all of which came back negative. I've tried the following meds (prescribed by doctors) - Ventolin inhaler, antibiotics (twice, don'-t remember the name of them), Nasonex for PND. Please let us know if you get a deffinitive diagnosis.
It is difficult to make a diagnosis on the basis of this small amount of information. If this is a new problem, it could be as a result of an infection. Wheezing and coughing are common symptoms of asthma. It is hard to understand the change in the sides of the symptoms you have described. Because these are clearly not normal symptoms, I would suggest speaking with your physician about these symptoms. It is likely that you will need a chest x-ray and a lung function test to help get a better understanding of the cause of the problem.
This is just mucous that has "settled" while you slept. In and of itself, it is not harmful, neither does it indicate anything serious. You may have allergies. If it has been more than 3 years since you bought a new pillow, you could try replacing it. It may befilled with dead skin and dust mites. Almost everyone is allergic to dust mite poo. You can also try drinking a lot more water. This will thin your mucous, and maybe keep it from settling at night.