Is what a mother does, collecting money from her children and buying gold for them without her husband's knowledge, considered haram (forbidden)? And is Zakat obligatory on this piece of gold?

It is permissible for a father to give money to his young child to buy his usual necessities, and it is not permissible for the mother to take it and dispose of it by selling. Rather, she should guide her child's spending and return the surplus to the father because he is the legal guardian of his money. The father is the one who preserves and develops the money given to the children, and he may take from it if he needs to. If the mother knows that the father takes his children's money and spends it while he is rich and does not preserve it for them, she may keep the surplus beyond their needs until they reach maturity and then hand it over to them; because in this case, the father is negligent and transgressing their rights. However, she does not have the right to dispose of it by selling or similar actions. The majority of jurists believe that the mother has no guardianship over the minor's money. Therefore, it is not necessary to inform the father about the money given to the children as long as he does not manage it well.