Is it necessary to repeat the prayer if something—most likely not a wind—exited during ablution or prayer, or is the action valid by continuing the prayer in that state?

If you are not certain that a f-rt has been released, then your ablution (wudu') is valid, and you are not required to repeat the prayers. The Prophet, peace and blessings be upon him, was asked about a man who imagined he felt something during prayer, and he said: "He should not leave until he hears a sound or finds a smell." However, if you are absolutely certain that something has been released from the anus, even if it is unusual, then your ablution is nullified according to the majority of scholars, with the exception of the Malikis.