What is the ruling on performing the prayer and reciting the Qur’an upon discovering the exit of stool remnants after performing ablution (wudu’), and after completing the prayer and Qur’an recitation, and is it obligatory to re-recite what was recited?

What you observed of stool after the prayer, if you are unsure about the time of its exit, then act as though it exited at the time you became aware of it, and you are not required to repeat the prayer. However, if you are certain that it exited before the prayer and you prayed unaware of it, then you are not required to repeat [the prayer] according to the more correct of the two scholarly opinions. As for reciting the Quran, you are not required to repeat it, because ritual purity is not a condition for it unless you are in a state of major ritual impurity (janabah (definition)), in which case ghusl (definition) (ritual bath) is obligatory before recitation.