Slipped Capital Femoral Epiphysis (SCFE): This condition is the most likely diagnosis given the patient's age, symptoms, and physical exam findings. SCFE is a common cause of hip pain in adolescents, particularly those who are overweight, and is characterized by the displacement of the femoral head from the femoral neck through the physeal plate. The patient's short stature and increased weight, along with the external rotation and shortening of the left lower extremity, are consistent with SCFE.

Legg-Calve-Perthes disease: This condition is another possible cause of hip pain in adolescents, characterized by avascular necrosis of the femoral head. It typically presents with hip pain, limp, and limited range of motion.
Transient synovitis: This is a self-limiting condition that can cause hip pain and limited mobility, often following a viral illness.
Apophysitis (e.g., iliac apophysitis): Inflammation of the apophysis, which is a growth plate, can cause pain in the hip and groin area.

Septic arthritis: Although less likely, septic arthritis is a medical emergency that requires prompt diagnosis and treatment to prevent long-term joint damage and potential systemic complications.
Bone tumor (e.g., osteosarcoma, Ewing's sarcoma): Although rare, bone tumors can present with hip pain and should be considered in the differential diagnosis to avoid delayed diagnosis.
Inflammatory arthritis (e.g., juvenile idiopathic arthritis): This condition can cause hip pain and stiffness, and early diagnosis is crucial to prevent long-term joint damage.

Gaucher's disease: A rare genetic disorder that can cause bone pain and hip problems due to the accumulation of glucocerebroside in the bones.
Multiple epiphyseal dysplasia: A rare genetic disorder that affects the development of the epiphyses, leading to joint pain and stiffness.
Sickle cell disease with hip crisis: Although rare, sickle cell disease can cause hip pain and crisis due to vaso-occlusive episodes in the bone.