What causes trachea spasms (which blocks air passage) whenever coughing occurs? Is that an asthma symptom?

Age dependent. The trachea of younger kids is somewhat rubbery, like a soft silicone or thin walled plastic. It does not spasm, like the soft walls of the gut, but it can and does vibrate enough to produce some odd sounds ( the croup noise) common from ~6mo-6yr. In tracheo-malacia, ( a transient condition) the walls are much weaker and actually cave in on each other, but soft walls not spasm is the problem.
Not possible. Your trachea is a rigid tube, it cannot spasm.