Is it permissible for the wife's mother to enter her daughter's house without permission, having a key, and is it permissible for the husband to compel his wife to allow this, and what is the proper way to deal with this situation?

The wife's mother does not have the right to enter suddenly without permission, and the husband is not permitted to agree to that. Rather, he must advise her with wisdom and good counsel, because honoring parents does not involve causing harm to the wife. The wife should be patient and not allow Satan to sow discord between them. She can seek the help of close relatives to pray for him and his mother to be guided.