One thing you could start with:
regarding “Pope Joan”- historical fact. Reference current events, ask them if the priest scandal got a lot of attention? Ask them to recount how the media and people in general reacted to this…Catholics alike. They will, no doubt, describe the circus that followed and is still going on.
Then ask them to explain why, exactly, it too 400 YEARS for the first account to come about of this alleged incident? Wouldn’t a POPE giving BIRTH, which is a very noisy, messy occastion, cause a great deal of consternation and hullabaloo? Worthy of several eyewitness accounts? Worth of ON THE SPOT “reporting” from that very time period?
These people do not have a grasp of logic, so you have to force them to use it. Likely they’ll still bury their heads in the sand, but that makes it easier to brush the sand from your own hands and feet and walk away.
The “Whore of Babylon” thing? Yeah…I have a friend who is a Protestant Evangelical Pastor. He says that those who refer to the Catholic Church as the “Whore of Babylon” have clearly not read Revelation and spend more time criticizing others then they do trying to follow in the footsteps of Jesus.
Others here will give you a great answer,on that, but regarding the former, do some research and you will find an abundance of logic to contradict this myth.
Just remember to keep your cool when talking to them. That will drive them nuts. Imagine Jesus is standing next to you while you have these conversations and act accordingly.
Good luck!
If all else fails, tell them that the “Whore of Babylon” is your mother and see if they open and close their mouths like fishes while they try to think up a response to such an outrageous comment.