Is it permissible for a mother to give her daughter gold in return for her managing household expenses, and do her siblings have a share in it? Is she sinful for shunning her son who does not provide for her? And is it permissible for her to give her other son gold to help him with marriage and studies?

There is no harm in a mother favoring some of her children over others in giving gifts, if there is a legitimate justification for it, such as one of them being poor, ill, in need of marriage, or seeking knowledge. So, if what your mother spent on you from gold was to help you with the expenses for her, your father, and your brother, or if what she spent on your brother was to help him with marriage, then that is permissible and there is no harm in it, due to the presence of a legitimate reason and justification.