What is the ruling of the Sharia on informing a woman's family or the competent authorities about a non-mahram man who entered her house at night, given that three witnesses saw the incident and confirmed that he is not one of her mahrams? Is this considered qadhf (slanderous accusation) that requires four witnesses? And what are the opinions of the four madhhabs on this issue?

If you are certain that a man has unlawfully entered upon a woman, leading to a private seclusion (khalwa) or similar situation, and he is not legally married to her, then you must advise her to fear Allah and repent from this forbidden relationship. It is preferable that the advice be given by trustworthy women.

If she does not respond, then you must escalate the matter to her guardian (wali) or one of her mahrams who is capable of deterring her from this wrongdoing, while adhering to wisdom and patience. This is not considered slander (qadhf), for qadhf is an accusation of committing indecency, and you are speaking about a foreign man entering upon her, which does not necessarily imply indecency. Furthermore, four witnesses are not required, as that is specific to bearing witness to adultery (zina).