Is a girl considered sinful, and must she make up all the prayers she missed, thinking that the brown discharge that precedes menstruation is menstrual blood?

Ease your mind, for the matter is simple. Many scholars consider yellowish and brownish discharges (ṣufrah and kudrah) to be menstruation absolutely, as long as they occur within the period when menstruation is possible. Although we do not issue fatwas according to this opinion, there is no sin upon one who acts upon it and follows it. Even if we assume this opinion is weak, many scholars have permitted issuing a fatwa (definition) according to a weak opinion after an action has occurred and rectifying it would be difficult. Therefore, there is nothing upon you for what has passed, and in the future, you can follow any scholar you trust.