I would say that your DS is probably a little bit young to be diagnosed with Asthma - these days we tend to wait until 3 or 4 before we say for sure that little ones have asthma. Your Dr was right about the temperatures thing, they are a lot sicker with bronchiolitis than they are with asthma.

There are various sub-types and degrees of asthma; if there is a family history there, it is possible he will develop asthma himself, but not necessarily as bad as yours has been. some children are fine 90% of the time, but get wheezy when they are sick; others are fine all of the time except when they exercise (that was me as a child!) Medications to control it are a lot better than they used to be as well.

Good luck.