Is a woman's fast valid if a reddish-brown discharge appeared, which she thought was her menstrual period, and based on that, she did not pray Fajr, but then it later became clear that it was not menstrual blood, so she made up the Fajr prayer?

Scholarly matters are subject to the rulings of scholars. If this discharge is blood, it is menstrual blood as long as it occurs within a time frame where menstruation is possible. Fasting and prayer must be abandoned until menstruation ceases and purity is observed, either by dryness or by the white discharge. Therefore, the fast of that day is invalid, and prayer was not obligatory. You must wait until you are certain of observing purity.