Will giving Budecort and Montelukast to a child cause any side-effects?

Hello doctor,

My son is 3.5 years old. We live in a place which has a lot of dust along with cold weather. My son has been suffering from regular cold cough problems since he turned 1, on an average three times in a month. Sometimes, the gap is longer when we are too careful and maybe the weather is stable. Starts with a runny nose, then cough, coughing is too much when he lies down, at night and early morning. Many times he also gets fever.

We have visited several doctors and have been told it is RAD, we still do not know what is he allergic to, could be dust. In the past 2.5 years, he has regularly (when suffering) taken Allegra/Cetrizine, Nasivion drops, finished several bottles of Beta2, Salbutamol, Asthaline syrups, Paracetamol, sometimes but less of Azithromycin when it is throat infection, once or twice taken Kidpred also, also he has taken Montelukast 4 mg for certain intervals like taken for a few months and stopped and again taken.

Three weeks back he got cough, we went to pediatric pulmonologist and were advised to start Budecort 200 inhaler twice a day for a year. Also, Asthaline inhaler three times a day for a week but still has cough. Allegra for a week along with Montair 4 mg. We did not start Budecort but started with Asthaline inhaler, Montelukast and Allegra. It improved and third week again he has got cold and cough but intensity is little less.

My question is should we start Budecort 200 inhaler for a year and stop Montelukast? Or should we not start Budecort right now and continue with Montelukast for some time and check if it works? Or should we give both Budecort and Montelukast? Also is it fine to start with Asthaline inhalers whenever he gets cough? Lastly is there any way which we can find out what things he is allergic to? Or a list of common allergies?