Hello..It's not normal. Chronic cough lasting more than 2-3 weeks is always a case of concern.It needs to be investigated property and thoroughly.Consult a Postgraduate Physician and get your physical examination done.All basic blood tests as well as Chest X-ray, X-ray of Sinuses, ECG (as you said you have breathing difficulty and Fainting episode), sputum examination needs to be done earliest
Next Steps
Consult a Physician postgraduate in Medicine initially to get all above tests done and for proper physical examination and in particular respiratory and cardiac examination.
Health Tips
Rest and Protein rich full diet is advisable .Avoid exercise for few weeks .