Can a child's enlarged tonsils contribute to cough variant asthma/general allergies?

Probably yes. A child who has big tonsils in the throat and big adenoids, with bad allergy symptoms, likely coughs on and off, has trouble breathing through the nose, and at night has trouble sleeping due to snoring. Taking the tonsils and adenoids out, along with using steroid nasal sprays in the nose, may make his life a great deal smoother (sleeps better, coughs less, breathes easier through the nose).
Absolutely. You have a large impediment in the airway which can make snoring, coughing, post-nasal drainage worse....Then the child snores at night or wheezes with activity. I am a proponent for tonsillectomy if the asthma specialist is unsure if it is the tonsils causing the respiratory problems or the asthma. Could also be both.