I, too, have had a nagging cough for over three months now. I have been on Amoxicillin, Ceftzil, and Azythromycin (Z-Pack). I have also tried Albuterol inhalers, nebulizer treatments, and Singulair tablets daily. I as well cannot shake the cough. It is very dry and raspy-sounding. My doctor and I are ruling things out one by one. Hang in there, they'll eventually find what the problem is!
You have asked lots of questions that are based on the assumption that this is, indeed, reactive airways disease. From the description you have given, an infection is the most likely cause of the ongoing problems. This could actually be from an infection of the sinuses and not the lower airways. Both can lead to exactly what you are describing. Also, without a breathing test it is not clear that this is indeed due to reactive airways from your lungs.
However, if the diagnosis is correct, no one can tell you how long you will need the inhaled steroids, weeks, months or years. Most of the time it is weeks, but it could be longer. Inhaled steroids in appropriate doses are safe and much less likely to lead to side effects than oral medicines. I would see how you are doing in the next week or so and speak with your doctor. If you are fine, he/she might stop the medicines. If not, you may need other tests, such as a CT of the sinuses and a pulmonary function tests (PFTs).