Is it permissible for a son to receive zakat from his father if the son does not have a stable income, and can this zakat be used to purchase gold to secure the educational future of the children?

If you are married and your husband is able to provide for you, it is not permissible for you to take from your father's zakat, because your maintenance is obligatory upon your husband, and the maintenance of your children is upon their father. However, if you do not find maintenance, then it is obligatory upon your father, and it is not sufficient for him to pay his zakat to you.