What is the ruling on performing Umrah and Salah with continuous flatulence, and is it obligatory to repeat them or offer expiation if the condition improves later?

If your feeling of passing gas is merely a waswasah (whispering from Satan), then pay no attention to it. However, if you are certain that gas was released, then you are considered to be afflicted with incontinence (sals). In this case, your 'Umrah is valid and you are not required to repeat it. Furthermore, the view that purification (taharah) is not a condition for Tawaf is a strong one, chosen by a group of verified scholars, and there is no blame upon you for adopting it when needed. Thus, your 'Umrah is valid, your marriage is valid, and no fidyah (expiatory payment) is due from you.