What is the ruling of Sharia on how to handle expenses for children? Am I obligated to cover the expenses of my younger son to match the expenses of his older brother, taking into consideration the presence of another daughter and her education expenses? And what is the ruling on zakat for the gold owned by me and my daughter?

Expenditure on children is based on their needs, and fairness is not obligatory in it. So, if the schooling of one child costs a greater amount, the other child is not required to be given an amount equal to that excess. As for jewelry prepared for use, Zakat (definition) is not obligatory on it according to the majority of scholars.