colder air tends to be drier, because theres little to no evaporation, plus you also stop sweating which is another factor. atopic dermatitis is pretty bad during late winter and when it ends. cold temperature also constricts your bronchial tubes slightly too, if you allergies, or ashtmatic its worst, so that induces coughing more. warm temperature would cause your lungs/tubes to dilate so to allow more air in. also the fact that allergies, cold/flu and even covid causes your lung to produce thick sputum which triggers the cough response.

normal sputum from allergie sis clear. while infections is thick and white, or yellow midly brown, sometimes tinge of blood. alot of green means it could be bacterial pneumonia.