Is it permissible to kill the paternal uncle who sexually and physically assaulted the mother, and whom the father and grandmother did not believe, and is it permissible to seek revenge on him?

You are not permitted to kill your uncle for the reasons mentioned. You must fear Allah and abandon these thoughts, for what your uncle committed does not warrant killing. Killing a soul unjustly is one of the gravest sins. Allah, the Exalted, says: "And whoever kills a believer intentionally, his recompense is Hell, wherein he will abide eternally, and Allah has become angry with him and has cursed him and has prepared for him a great punishment." [An-Nisa: 93].

If you witnessed what happened, then bear witness with your mother before your father and grandmother, as long as she has informed them. All of you must advise this man and call him to repentance. If he repents, then praise be to Allah. Otherwise, he should be shunned as a rebuke for his vile act and to avert his evil.