Are these whisperings—of doubt concerning the pronunciation of letters in prayer, repeating the Tashahhud, feeling God's displeasure, and matters of creed and God's existence—an illness or a devil's whisperings?

Waswas (obsessive whispering/thoughts) is a dangerous disease that corrupts one's religion and worldly life. It can also be psychological, and consulting doctors is recommended. Its only cure is to ignore it and pay it no mind. So, whenever Satan deludes the worshipper into thinking that he has not recited such-and-such, he should continue his prayer without concern. The same applies to waswas related to beliefs; one should not care about it or pay it any attention, for his detestation and repulsion towards it are proof of the sincerity of his faith.