The tunnel vision and lightheaded feeling aren't normal. Some coughing after a hard effort ("pursuiters' cough") is.
I suggest going to the doctor if you have a good one. You can get asthma tests, and they can (at least in some cases) tell if you have asthma even if they are not stressing you hard enough to have major symptoms.
I would up dragging my bike and a trainer to Standford to get tested- they wanted to test me on a treadmill but I can no longer run and certainly not hard enough to trigger my asthma symptoms. They had no problem with me bringing in the bike- the guy performing the test was a cyclist.