Is it permissible to stay overnight with female relatives who do not fully observe Ḥijāb while lowering the gaze, and to eat with them under the same condition? What is the ruling on inviting a male relative who might shake hands with the mother?

There is no harm in visiting relatives and staying with them for a full day, provided that Islamic guidelines are adhered to, such as avoiding unlawful mixing (ikhtilat) or seclusion (khalwa). It is permissible to eat at the same table if Islamic guidelines are observed; otherwise, it is not permissible. If going to them leads to falling into a forbidden (haram) act, then it is not permissible to go to them, and they can be kept in touch with through other means. If it is highly probable that inviting a relative to visit the house will lead to them committing a sin, such as unlawful handshaking, then the invitation is not permissible. One should be a reason for teaching them matters of their religion with wisdom and good admonition, and by acting upon what one calls them to.