Is it permissible to take the money remaining from the expenses of a sick brother's treatment and funeral after his death, or must it be distributed?

The intention of the one who donates money for equipping the deceased is considered. If their intention was to equip the funeral and not to make a general donation, then any remaining money must be returned to them, unless they permit you to keep it. If they permit it and you (or you and your siblings) take possession of the money, then it is to be shared equally among you. The intention is known either by explicit declaration or by circumstantial evidence. Often, the donor intends to generally assist the family of the deceased.