What is the ruling on frequent flatulence for someone with irritable bowel syndrome, and what is the ruling on someone who repeats wudu and prayer many times due to doubt about something having exited from him, or about having forgotten the takbirat al-ihram or the taslim?

What you are suffering from is obsessive-compulsive disorder (waswasah), and its cure is to ignore it. So, when you relieve yourself, perform ablution (wudu') and do not repeat the ablution due to the obsession of something having exited. And when you begin the prayer with takbir, proceed with your prayer and do not repeat any of the actions of the prayer due to obsession. Your purification is not invalidated except by a firm certainty that something has exited, and ignore any doubt. If you are certain that wind (gases) has exited after purification, and it does not cease such that you cannot find time for purification and prayer, then your ruling is that of one suffering from incontinence (sahib al-salas). In this case, perform ablution once after the time for prayer begins, and pray with this ablution as much as you wish until that time expires, and whatever exits from you will not harm you.