Is it possible for a 14-year-old to have prostate cancer?

Rare, not impossible. Nothing is impossible 14 year old boys have a prostate, and any organ can get cancer it would be extremely rare and likely the source of a journal article in a patient that young, it would most likely be a rhabdomyosarcoma, not a traditional adenocarcinoma.
Not likely. Youngest patient i ever saw was nearly 40. If you haven't hit puberty, you cannot develop prostate cancer.