I'm not overweight. But I breathe heavily. Is something wrong with my lungs?

Or your heart. Breathing heavily may be a sign of lung disease, but can also be a sign of heart disease, and need to be evaluated by your doctor.
Unknown. In the absence of any information being provided, no one can make such a determination on an internet site. As always, if you have a concern, you should see you doctor and let him examine you rather than try to get answers about you on the internet, regardless of the site.