As you said the child often gets cough and cold with increased breathing it is very likely to be an allergic cough which is very common in children. Incidence decreases as the child grows. Most of them improves by the age 6-7. Those crosses this age will mostly improve by the age 12 years. History allergy is very likely to exist in family at present you also seems to be having allergies. But many times allergy history in family is not very significant.
Next Steps
You should consult a doctor because many things are needed to be examined.
Health Tips
Avoid dust, smoke perfumes, be alert with changing weather, Avoid pets , Avoid use of carpets etc