What is the ruling of the Sharia on marriage and independent living, given the mother's desire for her youngest son to remain with her in the family home, and the older brothers' desire for independence?

It is the wife's right to have an independent dwelling, and this independent dwelling can be a part of the family home. Try to convince your wife to live with you in a portion of the family home, so as to combine fulfilling her wish with caring for your mother. If she is not convinced and you are able to find a dwelling adjacent to your mother's home, then do so, so that you can help her and maintain ties with her. However, if the wife is not pleased and you are unable to provide that, then seek another wife.