Is it permissible for us—while living on the zakat our grandfather grants us—to accept zakat from someone else and save it for the future?

If you are eligible to receive zakat, it is permissible for you to take it from the person mentioned, whether you spend it directly or save it for your needs. As for taking zakat from your grandfather, it is not permissible because the expenses of the daughters' children are obligatory upon their grandfather (their mother's father), and giving them zakat would be an evasion of the obligation of maintenance. Therefore, he must spend on them and pay zakat to those who are eligible. It is obligatory for your grandfather to spend on the girls until they get married, and on the boys until they reach the age of maturity and puberty.