Question

Would the trachea collapse when you were exhaling if there were no rings of cartilage in its walls

Answer 1

Yes, the trachea would collapse in this situation.
The trachea is a tube composed of cartilage which connects the pharynx and larynx to the lungs. It is a critical structure for all air-breathing animals and its main responsibility is to facilitate the passage of air. The trachea is surrounded by several c-shaped and incomplete cartilaginous rings which prevent it from collapsing.

Answer 2

The answer is YES. The trachea would collapse during exhaling due to the negative pressure developed as a result of evacuating air from the lungs. Physics dictates that the area with high pressure (outside environment) pressure on the region with low air pressure (in the lungs during exhalation).