What is the ruling on killing a man for committing the crime of adultery and rape and entering the house by climbing over the wall, and would the brother be sinful for killing him? And what is the ruling on killing the woman if her complicity with him and her concealment of the matter are proven?

It is not permissible to kill the man, because adultery is only proven by confession or the testimony of four witnesses. Even if it is proven, only the ruler can carry out the punishments, in order to avoid chaos. Killing him will bring calamity upon the killer.

If you are certain of his crime, refer his case to the competent authorities with the evidence you possess.

It is also not permissible to kill the woman. If she was coerced, there is no sin upon her. If she was complicit, it is not permissible to kill her because that is the prerogative of the ruler. Her statement must be believed unless there is evidence to the contrary.