So, why is it that air that’s below freezing make some people cough when they suddenly breath it through their mouth? (my self included)
But if I breathe it through my nose, then it makes my nose and sinuses ache until I stop.
Could it be that air thats below freezing causes condensation or tiny ice-sickles to form on the back of the throat?
Eating ice cream also makes me cough. Curious about the answer myself
Condensation happens when warm humid air gets cooled to its dewpoint, causing some of the water vapor to condense on nearby cold surfaces. Clearly that’s not the issue: you’ve got cold, dry air moving over warm, wet surfaces.
In fact, I’ll suggest exactly the opposite to your theory: rapidly inhaling through your mouth results in cold, dry air being admitted to the lungs. It gets warmed up there and absorbs moisture from sensitive lung surfaces, causing them to become uncomfortably dry. Your body perceives this as the presence of a foreign irritant (i.e. it thinks you’ve inhaled some dust), and launches you into a coughing fit.
When you breathe through your nose, your sinuses warm and humidify the air before passing it along to the lungs; in fact, they’re really good at this. The result is that warm, moist air goes to your lungs, and your lungs don’t get dried out.
Cold air is generally drier that warmer air. Dry air irritates the lining of the throat and lungs, so…
It’s not the cold, it’s the humidity. 
Or lack thereof.