Is it permissible to use the money left by the father after his death to cover the expenses of the condolence gathering and burial, and then distribute the remainder among the heirs, with the male receiving the equivalent of two females’ shares?

This money is part of your father's estate, and it is not permissible to dispose of it before its division except with the consent of all heirs. Otherwise, each heir must be given their full share, based on the Prophet's (peace be upon him) saying: "The property of a Muslim person is not lawful unless given with his willing consent." As for "mourning expenses," if what is meant is gathering after burial and preparing food, this is forbidden and was considered by the Companions to be a form of wailing and a reprehensible innovation. Filial piety towards a parent after their death is achieved through supplicating for them, seeking forgiveness for them, honoring their friends, maintaining kinship ties that are only connected through them, and giving charity on their behalf.