How can the mother be saved and the children protected from the psychological problems resulting from the mother's actions, which include frequent going out, staying up late, and speaking with foreign men after the father's death, and what is the ruling of Sharia on the son's actions, which resemble abandonment of his mother because she feels her wrongdoing?

A mother has a great right to dutifulness and kindness, even if she errs and sins. Allah has commanded us to accompany disbelieving parents with kindness. This does not prevent advising her, warning her of Allah’s punishment, seeking help from righteous people, and preventing her from going out without a mahram, as this is a form of dutifulness towards her. Efforts can also be made to marry her to a righteous man who can protect her and ensure her chastity.