Will Allah forgive someone who committed the act of the people of Lut, then repented sincerely, or will he be tormented on the Day of Judgment and abide eternally in Hell?

Whoever commits an abomination before reaching puberty is not sinful, nor is the prescribed (Shar'i) punishment applied to him. There is no sin for which Allah does not accept repentance if the servant repents with sincere repentance, for despairing of Allah's mercy is a great sin, and it is among the major sins. Suicide is among the major sins, and Allah has forbidden Paradise to whoever kills himself, and he will abide eternally in Hellfire. The questioner should fear Allah and have a good opinion of Him, and not follow the footsteps of Satan.