Thanks for this thread, interesting. Here's my take.
Explain the key arguments for Catholicism to an atheist?
The best arguments for Christianity are not arguments. The best arguments for Christianity are actions.
As example, please consider this historical hypothetical...
Imagine that in all it's history, Christianity had not articulated any doctrine. Imagine that Christianity was expressed only in service to others. Imagine Christians had always focused all their time and energy on service, service, service. Nothing but service.
When people asked Christians why they serve, the best answer would be to shrug shyly, and then share a big smile and say, "Because it's fun!" And leave it at that. Leave it at that.
When people asked Christians what Christianity is, the best answer would to take them right on down to the homeless shelter, and hand them a mop.
Love is salvation. Not talk of love.
If Christianity had taken this public course, Christians would now be the most popular group of people on the planet bar none. There'd be tons of mystery, and everybody would want to know the inside secret.
Ideology, any ideology, is divisive by nature. The more ideology we do, the more divided we get. Case in point, Christianity, an ideology explicitly
based on love, is divided in about 47,000 directions. Divisions within divisions within divisions, leading to conflict, not love. That's a teaching.
I'm making a strong and perhaps challenging case here. I've putting my face in your face. Guess what the most likely result of my strong ideological statement will be?
Yep, you guessed it. Somebody slamming down the reply button to tell me I'm wrong, Wrong, WRONG! I do that very thing myself all the time. All the time. I'm doing it now.
Ideology breeds resistance, not persuasion, not conversion.
On the other hand...
If we absolutely insist on ideology and arguments, the arguments have to be offered within the atheist's own method. You have to use their method to make your case. Theological arguments are a complete waste of time, counter productive.