A burning sensation is not a symptom of asthma and unlikely to be a side effect of the inhaled steroids. Mucus can be a sign of asthma but should diminish as asthma symptoms come under control.
Heartburn is a common cause of burning in the chest. That the burning is not worsened with exertion makes it highly unlikely that it is coming from your heart.
If you have not been evaluated by a gastroenterologist for acid reflux, I recommend that it be your next step. A high percentage of asthmatics also have acid reflux. The discomfort you describe (sometimes burning, vague symptoms that seem like asthma, and occasional stabbing pain) sound as though they could be reflux. I have had asthma all my life and a few years ago found that my symptoms felt slightly different and my asthma medications were not relieving them. Acid reflux in an asthmatic can cause asthmatic-type symtoms rather than classic heartburn (stabbing pain.) It was not until an intern working alongside my allergist suggested reflux that I even considered it. I am now taking aciphex (although preferred nexium until my insurance company stopped covering it) as well as zantac to control the reflux. Both asthma and reflux become worse after a respiratory infection and no doctor (allergist, gastroenterologist or pulmonary specialist) can explain how they interact. I've searched the web and the answer is not there. After an infection (cold, flu, pneumonia), both worsen and the symptoms become problematic, but generally clear up over time. When healthy, both my asthma and reflux are under control, but it is a balancing act, especially given that they are treated by different specialists. I suggest you talk first with you allergist and ask if it could be reflux. If he/she does not recommend tests to determine whether it is, then it probably makes sense to visit a gastroenterologist for a second opinion. Best wishes for recovery.