Is what a person experiences — a desire to laugh upon hearing the word "Allah" or during prayer, repeating ablution and certain letters in prayer, and feeling cast out from Allah's mercy — an actual suffering from obsessive-compulsive disorder, or is it merely an illusion and a whisper [from Satan]?

The cure for obsessive thoughts (waswas) is to ignore them and pay them no mind. So, no matter how much Satan whispers about something exiting [from you] or about not pronouncing a letter, do not pay attention to it. Obsessive thoughts concerning creed do not cause harm as long as you dislike them, and you must struggle against yourself to get rid of them. You will be rewarded for that. What you have mentioned is not mockery or derision; rather, it is pure obsessive whispering.