Is it permissible for the one entrusted with distributing Zakat to give it to his impoverished cousin who is in debt, and then demand that he repay his debt and his mother's debt, or must he seek permission from the Zakat-payer (the one who appointed him)?

It is permissible to give zakat to your impoverished cousin and then ask him to repay the debt. There is no need to seek permission from the brother who is paying the zakat, because Allah the Almighty says: "Indeed, [prescribed] charitable donations are only to be given to the poor and the needy, and to those who work on [collecting] them, and to those whose hearts are to be reconciled, and to [free] captives, and to those in debt, and for the cause of Allah , and to the stranded traveler; [this is] an obligation from Allah. And Allah is all-knowing and wise."