- The Pope: This is probably the one that I have been having the most trouble with. Do Catholics believe that he is without error/sin? Or do they just believe that he can be without error in certain instances?
Hi and welcome!
I just wanted to pick this one up because I think it is so important.
For the question you asked, there are two words: impeccable and infallible. The first word means incapable of sin, and nobody uses it to refer to the Pope. There is no question of a Pope being unable to sin, as a brief look at history will demonstrate.
Infallable means incapable of error in teaching. The circumstances in which the Pope is infallible are very clearly limited, “The Roman Pontiff, head of the college of bishops, enjoys this infallibility in virtue of his office, when, as supreme pastor and teacher of all the faithful - who confirms his brethren in the faith - he proclaims by a definitive act a doctrine pertaining to faith or morals…” (CCC #891) This means several things:
- Anything the Pope said before he became Pope is not infallible.
- The Pope can certainly be in error about any facts. If we had a Pope who said the Earth was flat, it would be unfortunate, certainly, but it would be unrelated to papal infallibility and would not have to be believed by anyone.
- Things the Pope says about issues of judgement in a specific situation are pretty much excluded from infallibility.
- Anything the Pope says, even if it is a matter of faith and morals, in a private or unofficial communication, is not infallible.
What the Pope is infallible about is the doctrines and principles of faith and morals, when he acts officially in his capacity as the Pope.
Because we have had some really holy men as Pope for a long time, people can get confused. Our Pope is an extremely intelligent, learned, and holy man. I love reading his books (which he has written as a private person, not as the Pope), and I believe the insights contained in them to be true. I would probably follow what he said on many issues that are not covered by infallibility. But I would need to realize that in doing so, I would be following the guidance of a man, not the Holy Spirit. There is nothing wrong with following the guidance of a holy man, just so long as you don’t think you are following the guidance of God.
But that is one side of the Papacy. I’d like to mention also what is, to me, one of the huge benefits of Christ’s instituting the Papacy–Unity.
Christ said, "I do not pray for these [the apostles] only, but also for those who believe in me through their word, that they may all be one; even as thou, Father, art in me and I in thee, that they also may be in us, so that the world may believe that thou hast sent me.” (John 17:20-21)
There are really only two ways to have unity: authority and stasis. Meaning, you can one of three things:
- Authority to decide what is development in doctrine, and what is a departure from what was handed down from the Apostles (Catholic Church)
- Changelessness in most matters because nobody has that authority (Orthodox churches as I understand them, but my understanding is limited, so if I am wrong and any Orthodox are reading this, please don’t be offended)
- Fragmentation (Protestant churches)
If there is no authority, we generally end up voting. And as has been amply shown over the years in Protestant denominations, when you have a vote, and it is at all close, you don’t get unity, you get schism. The 40% that voted the other way go off on their own, and all is well until the next decision needs to be made. Sure, there is a price to be paid for unity in the Church. Sometimes the decision doesn’t go your way. But if you believe that the Church is under the guidance of the Holy Spirit, then it’s just possible that the Church is right and you are wrong. What a humbling feeling (in a good way)! 
In addition, there is the other thing Jesus mentioned: “… so that the world may believe…” The world can see visible unity under the Vicar of Christ (the Pope) a lot easier than it can see the invisible unity of all Christians. Both are important, but visible unity is a better symbol for the world.
Finally there is the fact that with an authority, there can be authoritative teachings. I can’t stand up and say the Catholic Church teaches whatever I want, because there are a verifiable set of teachings that can be consulted. Is it possible that some Catholics have spent so much time praying to Mary that they have neglected praying to God? Yes. Does that mean the Church says it’s OK just to pray to Mary and never to God directly? Absolutely not. Even if every Catholic in America decided to pray only to Mary and never to God directly, the teachings of the Church would remain the same. This is a great help to keep us from following the fads of the age. (The example above is not, in fact, a fad of this age, but I decided not to pick something more controversial.)
What’s true is true. There may be new ways to say the same truth, so people growing up in different environments understand it as well as their ancestors in the Faith did, but the Truth says true, and the authority of the Pope helps us to understand that.
One of the reasons I left the Church was because of (my misunderstanding of) Papal infallibility. It’s also one of the things I really appreciated about the Church when I came back. 
God bless you on your journey!
–Jen