A steroid inhaler for asthma like Pulmicort has very little systemic absorption and shouldn't concern you re: Lyme, etc.
Did the DR give you a rescue inhaler too, albuterol? This might be a better test of whether asthma is the source of your complaints.
I don't have a typical presentation with wheezing; I have cough variant asthma. That's not that uncommon though.
Have you been to a GI to consider gastric reflux?