Is it permissible to pray behind a perpetrator of major sins, and is his Tahajjud prayer accepted while he insists on committing the major sin?

The imamate of a perpetrator of major sins is valid because the rule is that whoever's prayer is valid for himself, prayer behind him is valid. However, it is better to follow someone else. Furthermore, committing major sins does not affect the acceptance of deeds, with the exception of major polytheism (shirk akbar) with Allah the Almighty, or chronic alcoholism and persistence in it. In such cases, the acts of obedience performed by the sinner are accepted with a deficient acceptance, provided they possess faith in Allah the Almighty. The meaning of piety (taqwa) in His saying, "Indeed, Allah only accepts from the pious" is avoiding polytheism. So, whoever avoids it and whose deeds are sincere in intention, their deeds are accepted.