What is the ruling regarding the prayers performed by the questioner while wearing a head covering that had small openings at the front due to loosened threads, given that the appearance of these openings varied at times? And how many prayers must be redone?

As long as you are not certain that your hair was uncovered during prayer, your prayer is valid. Doubt concerning a condition of worship after its completion does not affect it. Even if you were certain of the uncovering, your prayer is still valid as long as the opening was small, because the unintentional uncovering of a slight amount of 'awrah does not invalidate the prayer according to the majority of jurists.