your son has acute infection which has increased his symptoms looking at his coughing i would say there are chances that he has broncho constriction which is causing breathing problem and excessive coughing. for his infection let him take antibiotics but for his cough your doctor may have to start him with a broncho dilator inhaler or at least tablets.
now coming to the allergy
part , treatment has to be in step wise manner
first if his symptoms is for few months a year like 2-3 months then you start an antihistamine which he has to take daily at night. if your symptoms still persist then your start using steroid nasal spray
, it has no side effect and it controls the allergy very well and you should start 2 weeks before his symptoms start. if this also does not cure you.
then get yourself tested for allergy, it is called skin prick test you will know what you are allergic to, then if possible you can avoid the allergen or you can start treatment called immunotherpy. this is currently the best treatment available and your symptoms will easily reduce by 80-90 percent kindly visit your ent specialist
he will guide you further and immunologist he will do skin prick test and immunotherapy