(1) Does the LDS Prophet claim the Petrine Seat?
The simple answer to this quesion is "yes," but that's not quite right. The LDS Church recognizes Peter as an apostle of Jesus Christ and as head of the primitive church by virtue of receiving the keys of the sealing authority (Matt 16:19).
However, we believe that the primitive church came to an end in a general apostasy. Jesus taught, "there needs be a falling away first," (2 Thes 2:3) and that falling away occured soon after the deaths of the apostles.
From an article "Simeon Peter" in the Feb. 1975 LDS publication Ensign we read:
This commission [receiving the sealing keys] did not create the first pope, for Peter never called himself bishop of Rome. ... Peter’s successors were not Bible scholars, ministers, evangelists, bishops of any kind, or even general councils of the church. Peter was an apostle raised to presidency and a prophet and revelator for the church. No one less can perpetuate his office in any Christian organization.
So, we believe the primitive church (the church of the original apostles) came to an end and therefore there is no continuation of it. The Church of Jesus Christ of Latter-day Saints is a restitution or restoration of something lost (Acts 3:20-22). Part of that restoration process must include (and did include) the keys to the sealing authority as were given to Peter, as well the keys of prophecy and revelation.
Therefore, yes, the LDS prophet claims the "Petrine Seat" from the perspective that he must be a prophet and head of the faith just as Peter was a prophet and head of the faith as were the prophets before him — by restoration following a prophecied general apostacy.
(2) Does the Prophet have a College of Bishops/"Highest Teachers" in union with the Prophet which is supposed to aide him in teaching the Church as a whole?
Again, the simple answer is "yes." When you see official references to the Prophet, you will see him identified as "Prophet and President of...". This is because there are two roles. The Lord calls and sustains His prophet, but the apostles call and the membership sustains the president of the Church. This distinction is important for a later question, too.
The Prophet and two counsellors form the First Presidency of the Church (D&C 107:22). For a detailed description of how this happens, see my answer to this other question.
The Twelve Apostles are a quorum equal in authority to the First Presidency (D&C 107:23-24). This quorum's purpose (beyond testifying of Jesus Christ) is to officiate and regulate the affairs of the Church under the direction of the First Presidency (D&C 107:33).
The Quorum of the Seventy are a quorum equal in authority to the Quorum of the Twelve Apostles (D&C 107:25-26). This quorum's purpose is specifically missionary work (proselyting) (D&C 107:34).
Authority is only equal when the quorums are acting (in each quorum) unanimously (D&C 107:27-29).
These three quorums constitute the group in the LDS Church that has primary responsibility to aid the Prophet. Further...
The First Presidency has authority to organize the Church as necessary (D&C 88:74-75; 117-120). Over the decades presidencies have been formed to support our youth organizations, women's organization, children's organization, sunday school, and welfare efforts. These presidencies, too, council with and aid the Prophet.
Additional information about these offices may be found on the LDS Church's website.
(3) Does the Prophet have jurisdiction as universal pastor of all churches.
Yes. Only the Prophet holds all the keys of authority in the Church and it is within his authority to reorganize any council or congregation within the Church. (D&C 107:65-67; 132:7)
The Prophet is not a dictator. All things are done in the Church according to the Law of Common Consent (D&C 26:2 and see the question I posted about it).
The only example I can remember of Church members voting against a proposal from the First Presidency was when the Provo, UT Tabernacle burned down. The building was destroyed and the First Presidency's initial decision was to demolish the remains and sell the land. When this was presented to Church members in Provo, they refused the decision. This led to President Monson's later announcement that the building would be rebuilt as a temple.
(4) Is the Prophet immune from being dethroned from the office of Prophet unless it is his own decision to step down (Or can't he step down at all)?
The call of Prophet is a lifetime call. Certainly, he can refuse that call should he choose at any time (though none has). Though we hold people to standards for advancement in the Church (such as minimum worthiness requirements to enter our Temples), no one is compelled to anything in the LDS Church.
However, the Lord has provided for His people should a day ever come that the President chooses iniquity. Bishops are called as judges, and the presiding Bishopric has one judicial duty: to try the President of the Church. This is discussed in D&C 107:73-84. Specifically quoting from verse 84:
Thus, none shall be exempted from the justice and the laws of God....
You'll note I shifted from "Prophet" to "President." As I mentioned earlier, there are two roles. The congregation of the Church can reject their president, but only the Lord can reject His prophet.
Technically this process can happen in either of two ways: either the president is tried by the Presiding Bishopric or the general membership vote to disapprove his call (Common Consent). Given that today's membership is about 16 million, getting 8 million +1 to vote against is fairly unlikely.
If I remember correctly, no president other than Joseph Smith has been tried for his membership. Joseph was tried three times — all during the Zion's Camp march — and was exonerated all three times.
(5) Are Mormons encouraged to listen to the Prophet above any other figure in the hierarchy?
Yes. Frankly, this is an odd question. It's like suggesting a U.S. citizen should listen to a senator above the President... The last time any sizable group in the U.S. did that it led to our Civil War.
Nevertheless, remember what I said above about the Quorum of the Twelve Apostles and the Quorum of the Seventy having to act unanimously to be equal in authority to the Quorum of the First Presidency. Individual members may have their opinions, but the quorums do not act contrary to one another.
It should be remembered that the concept of command doesn't exist in the LDS Church. No Bishop can command his ward, nor can the Prophet command the Church. D&C 121:41-44 teaches that the Priesthood is to be used:
by persuasion, by long-suffering, by gentleness and meekness, and by love unfeigned; by kindness, and pure knowledge, which shall greatly enlarge the soul without hypocrisy, and without guile...
People are people and will upon occasion disagree, but by definition, if a disagreement cannot be politely resolved, something went very, very wrong. Ours is not a church of command.
Therefore, yes, we invite people to hear the Prophet's voice and obey, but we're not looking for slaves, or robots, or people who follow blindly. "...Men should be anxiously engaged in a good cause, and do many things of their own free will..." (D&C 58:26-29).
Like I said, kindof an odd question. Let's assume (as I obviously believe) that our current Prophet is, indeed, a living prophet of the living Jesus Christ. Why wouldn't you listen to him? Curious....
It should be noted that the primary purpose of the Law of Common Consent is to help people come to Christ. In rare occasions that's an expectation of clarification from the Prophet. In most cases, it's the member who needs to pray, study, ponder, and then pray to receive the testimony of the Spirit themselves. The LDS Church is not a democracy. We cannot tell the Lord what to do. The general membership does not have the right nor the authority to force, for example, social change upon the Church. Dictating to the Lord is a fast way to earn His wrath. the Lord's response to Habbakuk is a good example.
(6) If the Prophet can claim to speak infallibly...
The one and only person who is infallible is Jesus Christ. This is why we have the Law of Common Consent, so that issues can be brought before the Church (and, ultimately, the Lord) and resolved. We, as regular Church members, are invited to balance two teachings:
"Whether by my voice or the voice of my servant, it is the same" (D&C 1:38). When the Prophet speaks the word of the Lord, we are expected to listen.
"Study it out in your mind..." (D&C 9:8-9). We are expected to learn about and know the truth for ourselves.
Joseph Smith understood this basic balance when he taught that "a prophet [is] a prophet only when he [is] acting as such" (History of the Church, 5:265). And how do we know when he is "acting as such"? We study it, ponder it, review scripture, and then get on our knees and pray.
(7) Can the Prophet in union with the College claim to speak infallibly to declare new dogma?
"Infallible" is a big word. We believe the only infallible person is Jesus Christ, therefore the answer is specifically "no." This does not mean the doctrine isn't perfect and from the Lord. It only recognizes that imperfect Man is, indeed, imperfect. As I mentioned, this is why the Law of Common Consent exists. However, when the Prophet of the Lord in union with the Quorum of the Twelve and Quorum of the Seventy stand together to declare new dogma... maybe you should listen, study, review scripture, and get on your knees and pray.
- An example of this was the acceptance by the Church of D&C sections 137 and 138 as doctrine of the Church and Declarations #1 and #2. In all these cases, the Prophet didn't simply declare and we listen. The Church was invited to accept the additional scripture and changes in Church policy by vote in Solemn Assemblies. It's a big deal.
As a contrasting example, consider the unaffiliated "FLDS" (Fundamentalist LDS) church run by Warren Jeffs. He believes himself to be infallible, as does his congregation. The mess they're in is a very good reflection of what happens when you believe a mere mortal is equal with God, which is what an infallible person must be.
The Priesthood is the authority to act in the name of God — but only at the level of God if the bearer is clean and purified of all sin (D&C 50:26-28). And even then, we do not command. The true followers of Christ pursuade through patience and long-suffering because there has ever only been one perfect person: Jesus Christ.