Chronic intermittent dry cough with (and perhaps due to) itchiness of trachea. whose area is this - ent or respirology/pulmonology?

Both: I would first start with your primary care doctor. The cough could be a blood pressure pill, asthma, allergies, Whooping Cough, tracheal problems, etc. An ENT can scope your trachea whereas a pulmonologist can do this and look further down the trachea to the bronchial tree as well as perform pulmonary function studies. A chest X-Ray and even a Cat Scan can be ordered by any physician.