This is not an easy question to answer from a forum. It is possible that your son has asthma. It is also possible that the cough is not related to asthma. "Reactive Airways Disease" is usually also called asthma. However, it appears from your note that you have never heard wheezing from your son. If you have heard wheezing, that would make the diagnosis a lot easier. If you are using the albuterol and it is working, it is, again, another sign that this is asthma. If that is the case, you may want to speak with your pediatrician about using the Pulmicort more regularly and/or seeing a specialist. The specialist would confirm or exclude the diagnosis and work on an effective preventative program to avoid the albuterol on a regular basis.
I certainly understand your concern. You need to have this discussion with your child's doctor as to whether his condition is asthma or something else. My kids were both diagnosed with asthma as infants (they're now 12 & 14 & very active & athletic & still asthmatic) & I believe I was diagnosed with asthma as a toddler or preschooler. Please work closely with your doctor to help maximize your son's lung function. In particular, please be sure to discuss your son's frequently coughing up a lot of mucous and what options/alternatives he can suggest.
Aloha,
Starion--thriving in HI
Your son doesn't have to wheeze for him to have asthma. Not all asthmatics wheeze, some only have a wet cough. That is the type of asthma I have. There is an organization called mothers of asthmatics that can offer some practical help. They have a web site and local chapters. Children as young as 3 can be taught to do peak flows to moniter their asthma. I taught my three children that early.