Is the wife obligated to live with her husband's mother and serve her in the same house if she does not wish to do so due to problems, taking into account the mother's strictness and the husband's desire not to displease his mother?

The husband does not have the right to force his wife to live with his mother in the same place. The wife is not obligated to do so, nor is she obligated to serve her husband's mother. She has the right to refuse to live with his relatives due to the harm that may come to her from it. However, it is preferable for the wife to serve her husband's mother to the best of her ability, as this fosters affection, love, and a stable situation.