Is a girl's prayer valid if she performs ghusl after menstruation before seeing the white discharge, and the 'Asr athan occurs during the second rak'ah of her Dhuhr prayer? And is the Fajr prayer accepted if she finds a brownish discharge after finishing it, and she did not verify its presence before the prayer due to hardship?

The Dhuhr prayer is valid because one rak'ah of it occurred before Asr. The brownish discharge (kudra) is considered menstruation if it occurs within the usual period, otherwise not. If it is not considered menstruation, then ghusl (ritual bath) is not required; wudu (ablution) suffices, along with purifying the body and clothes. However, if it is considered menstruation, then ghusl after its cessation is valid. The Fajr prayer is valid if it is not certain that the brownish discharge appeared before it, and its appearance is estimated to have occurred after the prayer.