Is it permissible to be in seclusion with a strange man in cases of necessity to avoid problems, or what should a woman do in such situations, and is the husband's permission to go out obligatory in these cases?

The young man who has reached this age has become accountable (mukallaf). His viewing the woman as a mother figure has no bearing on the Shariah ruling. If the young man's entry into the house makes unlawful seclusion (khalwa) probable, then the woman was correct in avoiding seclusion. However, if the house is spacious and the young man can occupy a separate part of it, then he may be allowed to enter. The woman must seek her husband's permission when leaving the house, except in cases of necessity or if she knows her husband's consent. It would have been better to act patiently and deal with the problem wisely, far from impulsiveness, to avoid problems. For example, the woman could have allowed him to enter and then gone to her family's house herself.