Is it obligatory for someone who continuously suffers from abdominal bloating and the emission of sounds (involuntary flatulence) to renew their ablution if they feel a movement in their anus during prayer, or is it sufficient for them to perform one ablution for each prayer? And is it permissible for them to perform ablution before the Adhan to recite the Quran and then pray with the same ablution?

Merely hearing sounds or feeling movement in the abdomen does not mean that something has exited the anus. The default assumption is that nothing has exited unless there is absolute certainty. If you are certain that something has exited and it is continuous, such that it does not stop for a sufficient time for purification and prayer, then your ruling is that of one afflicted with perpetual impurity ( سلس). You should perform ablution (وضوء) after the entry of the prayer time and pray as much as you wish until the prayer time exits. The Maliki school of thought does not obligate you to perform ablution for every prayer in this situation unless you intentionally nullify your ablution. However, it is more cautious to follow the opinion of the majority unless there is hardship. But if what exits stops for a sufficient time for purification and prayer, then wait for that time, perform ablution, and pray, even if you miss the congregational prayer.