Does the stance taken by someone who joked about the Imam’s recitation constitute a mockery of religion that expels one from the fold of Islam, and does this entail making up that day's fast and the prayer that followed it?

Beware of obsessive thoughts regarding mockery of religion, for what you mentioned is not mockery of religion. Dismiss these obsessive thoughts. What occurred falls under the category of forbidden backbiting, so both you and your friend must repent from it.