Allergic cough. The respiratory system from the sinuses to the lungs is pretty limited in its natural, healthy response to irritation of any kind, including allergies or infection: you can cough, wheeze, swell up or make mucus. Coughing is not as common as the others but is possible. Other causes should be considered as well if it is peristent.
Yes. Yes it can, specially if you have post-nasal drip.