For question a, see Acts 15. Infallibility is not said to inhere in persons. It is attached, rather to certain sorts of acts in certain situations. Or that’s a way of looking at it.
If you want to tussle, I’m no longer interested. But the Catechism is online, and a great many notions are usefully (if nio persuasively — which is not the goal of the Catechism) explained. So if getting a notion of what we teach is your goal, I’d start there.
So James, the leader of the Jerusalem Council, the FIRST center for Christendom before Antioch, possessed infallibility when he uttered his final pronouncement after all of those present finished conferring?
Interesting...