Catholics, Protestants, the Pope
A 1: The simple response is that we call ourselves, and are, both Christian and Catholic. Christian is the name of the followers of Jesus. The Christian faith embraces many denominations of people who are believers in Jesus as Lord and Savior. Catholics are those followers of Jesus who recognize a special role of ministry for the Bishop of Rome, as the successor to Peter. As Bishop of Rome, he is also called the Pope and exercises a unique pastoral role over the whole Church.
However, it would be imperfect and inaccurate to suggest that "Catholic" is just another denomination. The word itself means universal. Early Church Fathers used the word to distinguish Christians united with the Bishop of Rome, from those who were sectarian. The sects existed in only some areas of the Christian world, while the Catholic Church existed universally among all the peoples who had heard and received the Gospel. At the time of the Great Schism between East and West in 1054, the Western Church retained the name Catholic, while the Eastern Church took the name Holy Orthodox. The Western Church was further split at the time of the Reformation with the Church in Rome retaining the name Catholic and the reformers using the name Protestant referring to their protesting abuses in Church life.
In more modern times, a new problem has arisen with churches that have broken from Rome and use the name Catholic, such as the Old Catholic Church. These churches are not in union with Rome and so are not Roman Catholic Churches. In some areas, especially urban, immigrant areas, they use the name deliberately to confuse and deceive people who might not know the difference. Most accurately, then, Catholics are those followers of Jesus who believe the Tradition and Teaching of the Church and accept the role of the Pope.
A 2: It is not just the authority of the Pope that makes us Catholic, though that is significant. As you may know, the word "catholic" means universal. Hidden within the word universal is a concept of unity. The Catholic Church cannot be universal if each church is independent of each other--even if that independence is kept in check by regions. We would then simply have multiple churches. The Bishop of Rome (the Pope) is a symbol of unity for the Catholic Church. There is a commonality, a bond that unites all Catholics around the world. While the Catholic Church shares a foundation of faith with other Christian churches, there are elements in the practice of our faith that differ from many other denominations, such as the hierarchical structure, the sacraments and some of our dogmas and doctrines.
A 3: There actually are no references to the Pope in the Bible. The term "pope" was coined in a later period of church history. The first thing to note is that Peter had a unique place among the Apostles. Certainly the gospels unequivocally attest to this. Peter is the only apostle who speaks for "the Twelve". When the other apostles speak, it is only on their own behalf. Peter is always mentioned first when the apostles are singled out, and it was to Peter that Jesus gave the keys of the kingdom. Peter was martyred in Rome in the 60's, and Catholics believe that the Bishop of Rome succeeds Peter in his role as leader of the Church.
The character, shape and role of the papacy have developed dramatically over the centuries. This should not be surprising, since we are a living church in a living world. As such it is subject to continual change as human history continues to unfold. Therefore, the current Pope is very different from Peter. One of the challenges of every pope is to find a way to make the structure of the papacy relevant for his generation, so that the pope may be at the service of the people of God.As for how God works in him, I imagine it is not much different than how God works in the rest of us, though perhaps God is more successful with the pope! As with anyone else, the pope is a sinner who needs to be redeemed by Jesus Christ. If a particular pope is sufficiently prayerful and holy, he may indeed be more receptive to God's will.
Q 4: What separates a Roman Catholic from a Protestant? I've always assumed that it involves additional sources of Doctrine (in addition to the Bible), such as Papal decrees, Ecumenical councils, etc.
A 4: It is simply not sufficient to lump all protestant groups together as we have been accustomed to doing. What are often called the mainline Protestant Churches, such as Lutheran, Episcopalian (Anglican), Methodist, Presbyterian, are in many ways similar to Catholics. None of our Churches read the Bible literally. We use the best of modern scholarship in seeking to understand what the Word of God is really saying to us. For Lutherans and Episcopalians the similarities are even greater, for we are all Liturgical churches with rituals that closely resemble each other. Still, while we emphasize what we have in common, there are differences that cannot easily be dismissed. Some of it is doctrinal, and some has to do with the authority of the Pope. Recall that the Reformation did not take place until the middle of the sixteenth century. Most of the Church councils took place before that time. So the doctrines set by the various Council Fathers are, for the most part, accepted by these other Churches as well. Still, on some issues, our understanding has diverged since the Reformation.The role of the Pope continues to be a problem for some Protestant Churches. Even those that have a hierarchical structure, do not necessarily want to give central authority to the Bishop of Rome, preferring to see him as, at most, first among equals among the world's bishops.
A 5: Branches is not a good designation since, the Church is not truly like a corporation. Initially there was only one Christian Faith, comprised of local Churches throughout the Roman Empire. When Peter was martyred in Rome, he was succeeded by a bishop, Linus, who assumed his authority and responsibility for the Roman Church. The successors that followed in Rome, eventually came to be called Popes. Early on, there was a consensus that the bishops of the different churches collectively were responsible for the truth of the faith, hence the calling of ecumenical councils when major theological disputes needed resolution. The bishop of Rome was first among equals. Things became complicated when the capital of the empire moved to Constantinople, modern day Istanbul. Who, then, should have pre-eminence, the bishop of Rome or the bishop of Constantinople? It was as much a political concern as a religious one--perhaps even more so. Throughout the centuries, there were a number of religiously-motivated but politically-oriented splits between the Church of the East under the leadership of the bishop of Constantinople and the Church of the West under the leadership of the Bishop of Rome. Finally in 1054 a final split took place giving rise the Orthodox Church led by the Patriarch of Constantinople and the Catholic Church led by the Bishop of Rome. The Orthodox Church further divided along ethnic lines. Over the course of time, a number of ethnic groups reunited with Rome accepting the authority and primacy of the Bishop of Rome. However, while united in Faith, they retained their own liturgies (rites) and disciplines, e.g. married priests. These are called the Eastern Catholic Churches of which there are 22.