Short answers, pulled from the story.
The testicle has two primary functions: producing sperm and secreting androgens, chiefly testosterone. Sperm production is controlled by follicle-stimulating hormone from the anterior pituitary and by testosterone produced within the gonads themselves.
Testicles are external because spermatogenesis operates best at temperatures slightly below core body temperature. Carl Richard Moore proposed this in 1926. Testicular temperature in humans is maintained at 34.4 degrees Celsius, and temperatures above 36.7 degrees Celsius impede sperm production.
Cryptorchidism is the failure of one or both testicles to descend into the scrotum before birth. It occurs in about 3% of full-term births. If the testes do not descend far enough on their own within the first half year of life, surgery is required because undescended testes carry increased risks of infertility and testicular cancer.
Testicular torsion occurs when the testicle twists on the spermatic cord and cuts off its blood supply; it is a medical emergency. De-torsion surgery performed within six hours of onset carries a 90% chance of saving the testicle.
The number of sperm produced is directly proportional to testicular volume. Larger testes contain more seminiferous tubules and Sertoli cells, resulting in more sperm per ejaculate. Normal adult testicular volume ranges from 15 to 25 cubic centimeters, with an average of 18 cubic centimeters per testis.
The right whale has the largest testes of any known animal, with each testis weighing around 500 kg (1,100 lb). Testicular size across mammals correlates with mating systems and the intensity of sperm competition.