Asthma usually shows up in childhood, not at 17, and maybe some of the doctors you saw assumed that previous doctors would have checked out the possibility of asthma already. The definitive test for asthma is called a methacholine challenge; basically you inhale a chemical that irritates your lungs, and if you're asthma-prone then you have trouble breathing (there is no physical activity involved).

It probably isn't a heart issue if you haven't always had it...those are usually congenital...but I could be wrong. (Also not a doctor). But it sounds very debilitating, and worth fixing.