Should a man admonish his mother who insulted his wife, due to major problems between them?

You must combine showing dutifulness to your mother with treating your wife well, and strive to mend the relationship between them. If your mother insults your wife, advise her gently and politely without harshness. And it is the wife's right to live in a separate dwelling, and she is not obliged to live with any of her husband's family.