Is it considered a sin to boycott one's mother kindly due to her actions of selling gifted gold and differentiating between her children, and is it considered a shortcoming not to buy her gold while continuing to spend on her?

There is no objection to singling out one's wife with a gift rather than one's mother, and this is not considered an act of disobedience (uquq). It is preferable to conceal the gift from the mother. It is desirable to give a gift to the mother to bring joy to her. What a mother saves for some of her children is permissible if there is a need for favoritism, such as poverty or illness; otherwise, it is not permissible, because justice among children is obligatory. It is obligatory for children to spend on their mother if she is in need. If she is well-off, it is not obligatory except as a matter of recommendation and to gladden her heart.