Is it permissible for someone who suffers from excessive flatulence to perform ablution for each prayer and pray immediately without delay, in order to avoid hardship and missing prayers or daily activities, and is the first prayer he performed before the time expired considered valid if he experienced flatulence again?

It seems you are afflicted with waswasah (whisperings/obsessive thoughts). It is obligatory for one afflicted with waswasah to disregard and cast aside these whisperings. If you doubt the emission of wind, then dismiss the doubt and base your action on the original state, which is that nothing has exited, until you achieve complete certainty of its emission. This will provide you with ample time for worship. What you have read from us is not an act of extremism but rather the ease upon which the religion of Islam is built. We say that whoever finds sufficient time to perform purification and prayer without the emission of hadath (impurity), then they are not excused and are obligated to pray with a valid ablution. However, if the matter is uncontrollable and the previous time has passed without the hadath ceasing, then they are excused; they perform ablution for the entry of the next prayer's time and then pray. And a Muslim must reconcile their worldly preoccupations with their religious obligations.