Is it obligatory for a girl to inspect her underwear after waking up to ensure no nocturnal emission has occurred? What is the ruling on her prayer if she finds a white discharge after praying, and is she required to repeat it? How can she deal with persistent intrusive thoughts regarding the emission of madhy (pre-seminal fluid) and its effect on her ablution and prayers?

One must disregard obsessive thoughts. There is no need to inspect for pre-seminal fluid (madhy (definition)) or vaginal discharge before or after prayer. It is sufficient to act upon the default principle, which is that nothing has exited.

If there is absolute certainty that something has exited to the outer vulva, then ablution (wudu) is nullified. If the discharge was pre-seminal fluid (madhy), then the body and clothes must be purified, and ablution performed for prayer. If it was from vaginal discharge, it is considered pure according to the more preponderant opinion, and only ablution is required.

If you are unsure about the type of discharge, you have the choice to assign it the ruling of whatever you wish. If you are unsure whether the discharge exited before or after prayer, attribute it to the latest possible time it could have exited, and estimate that it exited after the prayer. There is no need to repeat prayers merely due to a doubt about the nullification of purity.