Is the fast valid for a woman whose menstruation stopped at dawn on the sixth day, and she saw the white discharge and fasted, then a faint brown discharge in a very small quantity came out after sunset, continuing until just before dawn on the seventh day, or must she make up that day?

Your fast is valid if you saw the purity before the dawn of that day, for purity interspersed during menstruation is valid purity. The brownish discharge during the period of menstruation is considered menses, but if it occurs after the end of the usual period, it is not considered menses.