Is it obligatory to repeat prayers in which there was frequent doubt due to obsessive whispering, and how can one distinguish between true doubt and the whispering of Satan?

The cure for obsessive thoughts (waswas) is to ignore them and pay no attention to them. Whenever Satan whispers to you doubts about the number of prostrations (sajadat), or uncertainties about recitation (qira'ah) or pronouncing "Allahu Akbar" (takbir), consider it two prostrations, and that you have recited and pronounced "Allahu Akbar." This ignoring of the obsessive thoughts should continue until they disappear.