Is it permissible to allocate a portion of the regular Zakat to support a family solidarity fund designated to assist grandchildren with their education, marriage, projects, and medical treatments? And is Zakat obligatory on the funds in this account when its conditions are met?

There is no objection to donation boxes and cooperating in good deeds if the donation is voluntary. It is not permissible to pay Zakat to such boxes, because the condition of transferring ownership to the recipient is not met, and because the beneficiaries might be the children of the Zakat payer. As for the obligation of Zakat on the funds in the box, it depends on whether the funds are still owned by the donors. If so, then Zakat is due on them according to each person's share. However, if the funds have exited their ownership and become like an endowment (Waqf), then Zakat is not obligatory on them, and it is not permissible to pay Zakat to them.