Is the repentance of a young man who committed sodomy by penetration once accepted, and is he obligated to carry out a Sharia-prescribed punishment upon himself, such as execution or being thrown from a high place, for his repentance to be accepted, knowing that he still feels lust towards boys?

Since you have repented from the great sin, rejoice in the acceptance of your repentance, for Allah accepts the repentance of His servants and forgives evil deeds. As for your inclination towards children, you must strive against it and take the necessary measures to repel it. Regarding the matter of the prescribed punishment (hadd), conceal yourself with Allah's concealment and do not expose yourself, for self-concealment is obligatory. And whoever sins and then repents with sincere repentance, Allah obliterates every trace of it through repentance.