What are the Main Characteristics of Mormon Prophets?

With regard to your Element (1), it does not appear that the Petrine Seat is a Latter Day Saints [LDS] concept that concerns them. They have no respect for the Catholic Priesthood or papal claims of authority, as they say the early Church soon became corrupted. They believe the only legitimate spiritual authority on earth today lies with the LDS and is God-given. They claim that there are no legitimate Prophets or Priests of God outwith their religion.

With regard to your other Elements, they seem to show an assumption that the LDS view of prophethood might have parallels with the Catholic system. Or perhaps you are hoping they do, so that you can spot similarities and differences. This will hardly be possible unless you first grasp fundamental differences between the Catholic and the LDS systems of Church governance.

This is what the LDS Doctrine & Covenants states in 43.1-7, apparently a revelation given through Joseph Smith in February 1831 –

“1 O hearken, ye elders of my church, and give ear to the words which I shall speak unto you. 2 For behold, verily, verily, I say unto you, that ye have received a commandment for a law unto my church, through him whom I have appointed unto you to receive commandments and revelations from my hand. 3 And this ye shall know assuredly—that there is none other appointed unto you to receive commandments and revelations until he be taken, if he abide in me. 4 But verily, verily, I say unto you, that none else shall be appointed unto this gift except it be through him; for if it be taken from him he shall not have power except to appoint another in his stead. 5 And this shall be a law unto you, that ye receive not the teachings of any that shall come before you as revelations or commandments; 6 And this I give unto you that you may not be deceived, that you may know they are not of me. 7 For verily I say unto you, that he that is ordained of me shall come in at the gate and be ordained as I have told you before, to teach those revelations which you have received and shall receive through him whom I have appointed. Quoted from https://www.lds.org/scriptures/dc-testament/dc/43.1-7?lang=eng#p1

For the office of Prophet, see this official LDS link, https://www.lds.org/scriptures/gs/prophet?lang=eng&letter=P

This is a claimed revelation given through Joseph Smith on November 11, 1831. This section was associated with the organization of the Quorum of the Twelve in February and March 1835. The Prophet likely delivered it in the presence of those who were preparing to depart May 3, 1835, on their first quorum mission. The verses refer to the aforesaid Doctrine & Covenants, section 107.

1–6, There are two priesthoods: the Melchizedek and the Aaronic; 7–12, Those who hold the Melchizedek Priesthood have power to officiate in all offices in the Church; 13–17, The bishopric presides over the Aaronic Priesthood, which administers in outward ordinances; 18–20, The Melchizedek Priesthood holds the keys of all spiritual blessings; the Aaronic Priesthood holds the keys of the ministering of angels; 21–38, The First Presidency, the Twelve, and the Seventy constitute the presiding quorums, whose decisions are to be made in unity and righteousness (and this is the highest court in the LDS church.) See https://www.lds.org/scriptures/dc-testament/dc/107.91-92?lang=eng#p90

Here is an extract from the book below, in a chapter about the Mormon Priesthood and Joseph Smith's claims about it:

"Smith and Cowdery were not eligible for the Aaronic priesthood. Only Israelites of the tribe of Levi and of the house of Aaron could be ordained to the Aaronic priesthood. Even then, they would have to demonstrate that their genealogy was complete. The Melchizedek priesthood, claimed by the Mormons as their higher priesthood, is mentioned in Smith's story:

'The messenger who visited us on this occasion and conferred this Priesthood upon us, said that his name was John, the same that is called John the Baptist in the New Testament, and that he acted under the direction of Peter, James and John, who held the keys of the Priesthood of Melchizedek, which Priesthood, he said, would in due time be conferred on us, and that I should be called the first Elder of the Church, and he (Oliver Cowdery) the second. It was on the fifteenth day of May, 1929, that we were ordained under the hand of this messenger, and baptized.' [Pearl of Great Price, Joseph Smith, 2:72]

"...There is no date given for the visit of Peter, James, and John to bestow the Melchizedek priesthood... "Peter, James, and John were chosen, in the Mormon scheme of priesthood, to ordain Joseph Smith and Oliver Cowdery to the Melchizedek priesthood. This, they claim, was done sometime between 1830 and 1833, with no place or time stated and with no witnesses present. The episode was written into an 1830 portion of 'The Doctrine and Covenants', Section 27. About 420 words were added to this section, possibly three years after the revelation was given, with no footnote explaining the addition.

"Joseph Smith needed documentation for his adoption of the name 'Melchizedek' for this higher priesthood. Therefore, in his 'Inspired Version' [of the KJV], he augmented the story of Melchizedek by about 470 words, calculated to enlighten us on the personal background of Melchizedek and to explain his priesthood from Adam to the present. Here is a sampling of the material added to the story in Genesis:

'... And thus, having been approved of God, [Melchizedek] was ordained an high priest after the order of the covenant which God made with Enoch [who was 'translated' into heaven without dying]. It being after the order of the Son of God... And men having this faith, coming up unto this order of God, were translated and taken up to heaven. And now, Melchizedek was a priest of this order.' ['Inspired Version', Joseph Smith, re. Genesis 14:25-33]” Quote from the book Is Mormonism Christian? Gordon H. Fraser, chapter 10, The Priesthood (Moody Press, Chicago, 1977 edition) ends.

See also the official LDS link, https://www.lds.org/scriptures/gs/translated-beings?lang=eng

This is the basis of LDS Church governance – the Melchizedek priesthood, with the office of the First Presidency, the Twelve and the Seventy being the highest Court in the LDS church. You may judge for yourself if there are any parallels or even similarities with the Catholic Pope and the LDS Prophet and their supporting officials.

1. The LDS believe the prophet holds the same keys and authority that Peter held.

As members of The Church of Jesus Christ of Latter-day Saints, we are blessed to be led by living prophets—inspired men called to speak for the Lord, as did Moses, Isaiah, Peter, Paul, Nephi, Mormon, and other prophets of the scriptures. We sustain the President of the Church as prophet, seer, and revelator—the only person on the earth who receives revelation to guide the entire Church.¹

2. The prophet is assisted by two counselors and a Quorum of Twelve Apostles.

We also sustain the counselors in the First Presidency and the members of the Quorum of the Twelve Apostles as prophets, seers, and revelators.¹

There are other General Authorities and General Auxiliary groups that help lead and teach the church:

The General Authorities consist of the First Presidency, the Quorum of the Twelve Apostles, the Presidency of the Seventy, the First and Second Quorums of the Seventy, and the Presiding Bishopric. (View General Authorities for more information on the roles of these leaders.)

The General Auxiliary presidencies consist of the Primary, Relief Society, Sunday School, Young Men, and Young Women General Presidencies.²

3. In a sense yes. If a General Authority visits a ward(local congregation) he presides(meaning to occupy the leading position of authority) and will often be given an opportunity to address the congregation:

The bishop oversees ward meetings. He presides at these meetings unless a member of the stake presidency, an Area Seventy, or a General Authority attends...Presiding authorities and visiting high councilors should be invited to sit on the stand.³

4. The prophet doesn't hold a throne and isn't king so there is no dethroning. The call to apostle or prophet is a life calling. Ezra Taft Benson said:

God knows all things, the end from the beginning, and no man becomes president of the church of Jesus Christ by accident, or remains there by chance, or is called home by happenstance⁴

Even prophets who become incapacitated with age or disease remain in their position, only those in the 70 or below will be released because of age or sickness. ⁵

5. The LDS church takes a political neutral stance (It is too long to quote here, read for details)

6. Mormonnewsroom lets us know that:

Not every statement made by a Church leader, past or present, necessarily constitutes doctrine. A single statement made by a single leader on a single occasion often represents a personal, though well-considered, opinion, but is not meant to be officially binding for the whole Church.

Or more cleary put by James E Faust:

We make no claim of infallibility or perfection in the prophets, seers, and revelators

However, members are encouraged to trust the living prophet and heed his counsel:

We can always trust the living prophets. Their teachings reflect the will of the Lord, who declared: “What I the Lord have spoken, I have spoken, and I excuse not myself; and though the heavens and the earth pass away, my word shall not pass away, but shall all be fulfilled, whether by mine own voice or by the voice of my servants, it is the same” (D&C 1:38).

Our greatest safety lies in strictly following the word of the Lord given through His prophets, particularly the current President of the Church. The Lord warns that those who ignore the words of the living prophets will fall (see D&C 1:14–16). He promises great blessings to those who follow the President of the Church:

“Thou shalt give heed unto all his words and commandments which he shall give unto you as he receiveth them, walking in all holiness before me;

“For his word ye shall receive, as if from mine own mouth, in all patience and faith.

“For by doing these things the gates of hell shall not prevail against you; yea, and the Lord God will disperse the powers of darkness from before you, and cause the heavens to shake for your good, and his name’s glory” (D&C 21:4–6).¹

7. Yes the prophet in union with the Twelve can declare new doctrine (though you won't find the word infallible used to describe any LDS doctrine). Mormonnewsroom continues:

With divine inspiration, the First Presidency (the prophet and his two counselors) and the Quorum of the Twelve Apostles (the second-highest governing body of the Church) counsel together to establish doctrine that is consistently proclaimed in official Church publications. This doctrine resides in the four “standard works” of scripture (the Holy Bible, the Book of Mormon, the Doctrine and Covenants and the Pearl of Great Price), official declarations and proclamations, and the Articles of Faith. Isolated statements are often taken out of context, leaving their original meaning distorted.

In quick answer on why you don't see the word infallible used to describe anything the LDS church teaches can be summarized from the Book of Mormon title page:

And now, if there are faults they are the mistakes of men; wherefore, condemn not the things of God, that ye may be found spotless at the judgment-seat of Christ.

_{1 https://www.lds.org/topics/prophets?lang=eng}

_{2 https://www.lds.org/topics/church-organization/how-the-church-is-organized?lang=eng&old=true and a list of the current leadership see https://www.lds.org/church/leaders?lang=eng}

_{3 https://www.lds.org/handbook/handbook-2-administering-the-church/meetings-in-the-church/18.2?lang=eng&_r=1#182}

_{4 https://www.lds.org/manual/teachings-of-the-living-prophets-student-manual/chapter-3?lang=eng}

_{5 article with examples http://www.mrm.org/lds-prophet-incapacitated}