The idea that Muslims worship the one true God is not a "private opinion of popes." It was the teaching of Vatican II in its document Nostra Aetate:

The Church regards with esteem also the Muslims. They adore the one God, living and subsisting in himself; merciful and all-powerful, the Creator of heaven and earth, who has spoken to men; they take pains to submit wholeheartedly to even his inscrutable decrees, just as Abraham, with whom the faith of Islam takes pleasure in linking itself, submitted to God. (NA 3)

The Catechism of the Catholic Church offers these explanations:

• During his public life Jesus not only forgave sins, but also made plain the effect of this forgiveness: he reintegrated forgiven sinners into the community of the People of God from which sin had alienated or even excluded them. A remarkable sign of this is the fact that Jesus receives sinners at his table, a gesture that expresses in an astonishing way both God’s forgiveness and the return to the bosom of the People of God. (CCC 1443)
• In imparting to his apostles his own power to forgive sins, the Lord also gives them the authority to reconcile sinners with the Church. This ecclesial dimension of their task is expressed most notably in Christ’s solemn words to Simon Peter: "I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." The office of binding and loosing which was given to Peter was also assigned to the college of the apostles united to its head. (CCC 1444)
• The words bind and loose mean: Whomever you exclude from your communion, will be excluded from communion with God; whomever you receive anew into your communion, God will welcome back into his. Reconciliation with the Church is inseparable from reconciliation with God. (CCC 1445)

When an ambassador is representing his country abroad, should he wear a T-shirt and jeans or formal attire? When taking part in official diplomatic ceremonies, are the marks of honor given the ambassador by his host country for him personally or for the country he represents? In like manner, the pope is the primary ambassador for Christ and for the heavenly kingdom. Formal attire honors the dignity of his office and the Person he represents. Marks of honor given the pope are not for him personally but for Jesus, who he represents.

The purpose of the Room of Tears is not to examine the pope’s body; it is to allow the new pope to change into papal vestments before receiving the pledges of obedience from his fellow cardinals and then being introduced to the people of Rome and to the world from the loggia above St. Peter’s square. The room is nicknamed the Room of Tears because new popes have often been overcome with emotion at thought of the heavy burden that has been given them. The story of Pope Joan is a legend, roundly dismissed by serious historians as unworthy of belief. (See "The Popess Who Just Won’t Go Away" in the January 2008 issue of This Rock)

The function of the pope in the New Testament is similar to, though not identical with, the function of the high priest in the Old Testament. For example, the high priest offered the greatest sacrifices to God (e.g., on Yom Kippur, the Day of Atonement) and functioned as the earthly leader of God’s people.

While Christ is our high priest with respect to sacrifices (he offered the great sacrifice of himself; see Hebrews 7-10) and is the ultimate head of all God’s people, he has left Peter here to serve as his visible representative in his absence. The pope, as the successor of Peter, thus has a function similar to that of the high priest as the earthly head of God’s people.

Papal infallibility means that the pope is protected from error when he "proclaims by a definitive act a doctrine pertaining to faith or morals" (CCC 891). This does not mean that he is impeccable (incapable of sin) or inerrant (incapable of error).

Although the quote is commonly attributed without source documentation to Pope Leo X, it is believed to have originated in a satirical piece titled "The Pageant of the Popes" by a Protestant controversialist named John Bale (1495–1563). Bale wrote: "For on a time when a Cardinall Bembus did move a question out of the Gospell, the Pope gave him a very contemptuous answer saying: ‘All ages can testifie enough howe profitable that fable of Christe hath ben to us and our companie.’"

The prayers you’re referring to are called the Leonine prayers after Pope Leo XIII, who asked they be recited after Low Mass for the intention of the whole Church. Later, Pope Pius XI requested they be said for the conversion of Russia. The prayers consisted of three Hail Marys, one Hail Holy Queen, a prayer to St. Michael the Archangel, and a three-fold invocation to the Sacred Heart.

These prayers were never part of the Roman Missal. The instruction to remove the prayers came before the Second Vatican Council closed in 1964.

The Church teaches that the "one flock, one shepherd" in John 10:16 subsists in the Catholic Church, governed by the successor of Peter and by the bishops in communion with him (Dominus Iesus 17).

While the Church acknowledges that many truths are found in other ecclesial communities, the complete fullness of Christ’s teaching and grace is only found in the Catholic Church:

This is why Jesus himself prayed at the hour of his Passion, and does not cease praying to his Father, for the unity of his disciples: "That they may all be one. As you, Father, are in me and I am in you, may they also be one in us, . . . so that the world may know that you have sent me." (John 17:21, CCC 820)

To work ceaselessly and efficaciously for the return of all humanity and all its goods, under Christ the Head in the unity of his Spirit is the mission of the Church (CCC 831).

With Pope Benedict VXI, "Let us remember it in our prayer to the Lord, as we plead with him: yes, Lord, remember your promise. Grant that we may be one flock and one shepherd! Do not allow your net to be torn, help us to be servants of unity!" (Homily of His Holiness Benedict XVI, St. Peter’s Square, April 24, 2005)

Further reading: Catechism of the Catholic Church, 11-870; One, Holy, Catholic and Apostolic: The Early Church Was the Catholic Church by Kenneth D. Whitehead

John Paul II considered his continuation in office to be an acceptance of the cross given him by Christ. He responded to suggestions that he resign by saying, "Christ didn’t come down off the cross." John Paul’s service to the end demonstrated to the world that the papacy is about more than efficiency. It is a call to service and a call to suffering. Not for nothing did Benedict XVI call the possibility of his own election "this guillotine" and pray to be spared it.