This is not an easy question and certainly might be answered in different ways by different specialists. Here is one opinion. With the history you are presenting it is likely, rather than a small chance, that this is indeed asthma. It may also be complicated by a chronic sinus condition. This is often silent, but looks like a cold before getting chest involvement. While allergy may play a role, with the history of prematurity, this is probably not a major factor, as much as the prematurity.
With respect to testing for asthma, this is not an easy procedure. It certainly is not done in the office of a primary care physician or a pediatrician. Lung function tests, even in 2-year-old children, can be performed in a special lab run by pediatric pulmonologists, usually in a university setting. We do this frequently here at National Jewish Medical and Research Center.
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In the mean time you may want to keep him inside in an aircondtioned room at all times when it is very humid outside and when the airquality is low, and when the pollen counts are high.
When I was little I had cancer and was always very sick and my parents said it helped out greartly keeping me in an airconditioned room during the summer months.