Is it Islamically permissible for a Muslim student to live with a Muslim family consisting of a divorced mother and three children, given that he has his own separate room and there is no khalwa (private seclusion)?

The basic ruling for traveling to non-Muslim countries for study and long-term residence is prohibition, unless the conditions for permissibility are met and one is safe from temptation. As for living with a foreign family in those countries, it is a condition that the room be completely separate with its own facilities; otherwise, it is not permissible. This is because it leads to unlawful seclusion (khalwa (definition)), which is a strong means to corruption and temptation. The Prophet, peace and blessings be upon him, forbade seclusion with a non-mahram (definition) woman, stating that Satan is the third party between them. The prohibition of khalwa is agreed upon, and this falls under the principle of sadd al-dhara'i' (blocking the means), which is the prohibition of something permissible if it leads to something forbidden.