I was diagnosed with asthma when I had a nagging cough that remained after all the other cold symptoms had faded by trying ventolin in a doctors surgery. Because that relieved my symptoms, I was told I had asthma.**
Try it. If it helps then maybe a visit to the GP is warranted. It could easily be asthma from how you describe it. For me the triggers are chest colds, allergens (particularly dust), exercise and cold dry air. My symptoms are so mild these days I need an inhaler less than once a year
**It was a revelation. The familiar symptoms of my childhood "chronic bronchitis" reoccured as an adult - tight chest, wheezing, an itch deep behind my breast bone - and were all relieved ventolin. It is now understood that children diagnosed with chronic bronchitis in fact were asthmatics.