This could be allergies or asthma, but I am no doctor. I am just giving examples of what could cause some of the symptoms you describe. If it is asthma, then that could explain why your heart rate goes up with running, from what you are saying, it could easily be that. Your chest probably feels constricted as well. Did your doctor by any chance do lung function tests? That might be a good idea along with post bronchodilator results (first they do the test and then repeat it after you have a treatment designed to open up your airways. If there is a significant improvement in your values post bronchodilator, then asthma seems likely)
I would definitely start there...get lung function tests. IF they come out completely normal, then I would ask for further testing, perhaps of the heart which can mimic respiratory disease for completeness sake.
If it does turn out to be asthma, then it would be a good idea to see what triggers it..such as allergy issues for example.
Hey I had a blood allergy test and everything was normal apparently I'm not allergic to anything. They also had me wear a holter monitor to see what's going on with my heart. Unfortunately there is a long queue of people waiting for the lung function test and that's scheduled to be done January 19th so that's a ways from now. So far the allergy test is normal and I think my heart test will show them plenty as I did all the things that trigger it to go fast and skip beats.
My throat is constantly dry and irritated though, I don't have a fever or anything just a dry cough and it feels like if I move my head around it irritates my breathing channel and makes me cough. Hopefully it's not asthma and it's a fungal infection or a virus or something that can be taken care of rather than something I have to deal with the rest of my life.