Is it permissible for a father to interfere with the privacy of his adult son living with him, or to search his belongings, or to prevent him from going out, or to ask him about his destination?

A father does not have custody over his adult, mature son. The son is allowed to live independently in his own dwelling and life. Ibn Qudamah said: "As for the adult, mature person: there is no custody over him, and he has the choice to reside with whichever of his parents he wishes. If he is a man, he may live independently." Ibn Uthaymeen said: "After maturity, he owns himself, and no one has authority over him, neither his father nor anyone else. However, if there is a fear of him falling into corruption, then his father must be given guardianship over him." Both the father and the son are advised to have good faith and deal kindly with each other.