I was recently in an accident, (no serious injuries) that caused the airbags to deploy. Immediately afterward there was a dust of some sort in the car of which I took several breaths. Since then, (about 30 hrs) I have been coughing fairly steadily, sometimes violently enough that I’ve had to sit down. My question is A. What is that mystery dust? and B. If this persists for much longer should I go the doctor/hospital?
-Droc
It’s rocket fuel. No, really. Airbags typically use sodium azide as a propellant, a solid rocket fuel. Saw it on the Bill Nye the Science guy episode they made me watch in Defensive Driving. 
The product of its combustion are nitrogen and sodium hydroxide (lye), but most of the dust from the airbag is from the cornstarch they use as a lubricant. Supposedly the amount of lye produced is very small. If your cough is violent and persistant I’d definitely say see a doctor. Are you allergic to cornstarch, perchance?
As far as I know I’m not allergic to cornstarch, didn’t know it was a lubricant either.
-Droc
The cornstarch isn’t a lubricant in the classic “slippery” sense, but serves to keep the airbag material from sticking to itself while packed into the cartridge. If the bag stuck together, on deployment, you’d have either a hard wad of fabric coming at you, or it would burst rather than inflate and not come at you.
Some airbags also use talc as a lubricant.
You may not be experiencing an algeric reaction, but rather the result of inhaling cornstarch / talc into your lungs.
FWIW not all air bags use sodium azide.