What is the ruling on giving zakat to an unmarried sister who has no financial means, and what is the ruling on spending this zakat on the children of the one giving it, such as for Eid?

There is no objection to giving zakat to a poor or needy sister, but it is not permissible to give her zakat on the condition that she spends it on the zakat payer's children, because their maintenance is obligatory upon him. However, if the sister spends the zakat on the zakat payer's children of her own accord after taking possession of it, then there is no objection to that.