Is a menstruating woman, who is reciting the Qur'an to others, obligated to continue her recitation if she sees a worshiper's 'awrah uncovered, or should she cover it? And does the same ruling apply if she is the one reciting to her teacher, or if she has to cross rows of worshipers to cover the 'awrah of one of them?

You did well by covering the 'awrah (private parts) of the girl that became uncovered during prayer. If, in the future, you see any part of your sister's 'awrah uncovered during prayer, you should alert her verbally. This falls under alerting the heedless and enjoining good, both of which are obligatory according to some scholars. Your recitation of the Quran should not prevent you from doing so, because it is permissible to interrupt Quran recitation for a legitimate need, such as enjoining good and forbidding evil. There is no harm in breaking the second row to alert a student in the first row, because stepping over people's shoulders for a benefit is permissible.