Are my sister's fasting and prayers acceptable, and is she sinful for touching the Quran, given the doubt about the completeness of her purification from menstruation due to yellowish and brownish discharges, and the presence of moisture after ghusl, which she doesn't know if it's yellowish or not?

Yellowish and brownish discharges are considered menstruation if they occur during the customary days, or if they are connected to blood, and they must have their specific characteristics and color, not merely be moisture. As for the moisture exiting the vulva, if it is neither yellowish nor brownish, it is not considered menstruation, and your ritual bath and fasting are valid; however, it nullifies ablution. If the moisture is continuous, perform ablution after the entry of each prayer time and pray as much as you wish. If the moisture exits definitively before or during the prayer, the prayer is invalidated, and ablution becomes obligatory. It is not permissible to touch the Qur’an except after performing ablution. If there is doubt about the time of its exit, the doubt should be referred to the last possible time. The sin of touching the Qur’an and praying after ablution is nullified by moisture depends on knowing that it is prohibited.