Asthma: This is the most likely diagnosis given the symptoms of shortness of breath with exercise, chest pain, and cough. Asthma is a common condition in adolescents, and exercise-induced asthma can present with these exact symptoms. The fact that symptoms improve once the exercise is stopped and the patient rests also supports this diagnosis.

Exercise-Induced Bronchospasm (EIB): Similar to asthma, EIB can cause shortness of breath, chest tightness, and cough with exercise. It's possible that the patient has EIB without underlying asthma.
Anxiety or Panic Disorder: Anxiety can cause shortness of breath and chest pain, especially in adolescents. The fact that the patient can push through the discomfort and that it resolves on its own could suggest a psychological component.
Vocal Cord Dysfunction: This condition can mimic asthma and cause shortness of breath and chest tightness with exercise due to abnormal vocal cord movement.

Pulmonary Embolism: Although rare in adolescents, a pulmonary embolism can cause sudden onset of shortness of breath and chest pain. It's crucial to consider this diagnosis, especially if there's a history of recent trauma, immobilization, or family history of clotting disorders.
Hypertrophic Cardiomyopathy: This heart condition can cause chest pain and shortness of breath with exercise due to obstructed blood flow. It's a leading cause of sudden cardiac death in young athletes, making it a critical diagnosis not to miss.
Pneumothorax: A spontaneous pneumothorax can cause sudden chest pain and shortness of breath. Although more common in tall, thin males, it should be considered in any patient with acute respiratory symptoms.

Cystic Fibrosis: Although less likely, cystic fibrosis can cause chronic respiratory symptoms, including shortness of breath with exercise. It's typically diagnosed earlier in life, but mild cases might not be identified until adolescence.
Primary Ciliary Dyskinesia: This rare genetic disorder affects the cilia in the respiratory tract, leading to chronic respiratory infections and symptoms. It could potentially cause exercise-induced symptoms, although it's much less common.
Mitral Valve Prolapse: This condition can cause chest pain and shortness of breath, especially with exercise. However, it's relatively rare and usually associated with other symptoms like palpitations or a murmur on physical examination.