What is the ruling on frequent flatulence when performing Umrah, and what should be done about the repeated breaking of ablution during circumambulation (Tawaf) and prayer in the Haram?

If your feeling of passing gas is merely a doubt, do not pay attention to it, and do not rule that something has exited except with certain conviction. If you are certain that gas has exited, but you find enough time during prayer to perform it with proper purification, then you are not afflicted with سلس (incontinence), and you must wait for the gas to stop, then pray and perform tawaf. There is no harm in missing congregational prayer in the Haram, and your prayer in the hotel is multiplied. Be diligent in performing tawaf while purified. However, if it becomes difficult for you, then you have latitude to adopt the opinion of those who do not stipulate purification for the validity of tawaf. This is the choice of Sheikh Al-Islam Ibn Taymiyyah, and it is permissible to avail oneself of some concessions when needed, which many scholars have made easy.