If you keep on exercising and getting asthma attacks, will your asthma attacks eventually go away?

No. You can't train away asthma. Some people with mild exercise-induced asthma continue to exercise when symptoms first appear. After a few minutes their symptoms resolve spontaneously. Most of us will not experience this. Instead asthma increases with prolonged exercise. Taking albuterol before exercise is the best 1st step. If not sufficient you may need a daily controller medication.
No. In fact the swelling in the small tubes of the lungs can get worse eventually leading to respiratory arrest.