What is the ruling regarding the prayers I performed while unaware of the presence of a dry semen stain on my sleeping garment, and I do not know the date of this nocturnal emission? Do I have to make up these prayers? If so, how many are there and how should I make them up?

If you find dry semen on a garment that no one else sleeps in, and you do not know when it occurred, then consider it to have happened during the last sleep you had in it. You must perform ghusl (ritual bath) and make up the prayers you performed after that last sleep, as they were without ritual purity. Their obligation is not fulfilled except by making them up.