Can Allah alleviate the torment for a person who committed the indecency of Zina with a girl from Ahl al-Bayt, and who is regretful and striving for repentance? And would establishing the حد (prescribed punishment) upon himself benefit him?

Adultery is a grave sin, but it does not, by itself, necessitate eternal damnation in Hell. Repentance from it is valid, just like repentance from other sins. It is not a condition for the repentance of an adulterer that the prescribed punishment (حد) be carried out upon him. Rather, repentance between him and Allah suffices, along with concealing his sin. Repentance involves ceasing the sin, regretting it, and resolving never to return to it. Sound repentance is accepted and obliterates what came before it. Allah rejoices in His servant's repentance and transforms his evil deeds into good deeds.