JOEL - First of all, the Journal of Discourses(JOD) has some problems with accuracy in terms of exactly what was said by Brigham Young and others. Speeches were recorded in shorthand and rewritten sometime later, often without giving the speaker the opportunity to proofread the speech. There was never any official revelation from God that was declared from within those discourses. They reflect the opinions and biases of the speakers as well as the bias and memory of them who recorded and later published them.

Quote #1. "I do not profess to be a Prophet. I never called myself so; but I actually believe I am, because people are all the time telling me that I am." (JOD 5:176)

Answer:
Brigham Young himself is not speaking here. This was a Discourse by President Heber C. Kimball, delivered in the Bowery, Great Salt Lake City, Sunday morning, August 23, 1857, which started on page 171 of volume 5 of JOD.

Quote #2. "A person was mentioned to-day who did not believe that Brigham Young was a Prophet, Seer, and Revelator. I wish to ask every member of this whole community, if they ever heard him profess to be a Prophet, Seer, and Revelator, as Joseph Smith was? . . . Who ordained me to be First President of this Church on earth? I answer, It is the choice of this people, and that is sufficient." (JOD, 6:319-320)

Answer:
Brigham Young is not saying he is not a prophet; he only said he was not a prophet like Joseph Smith was. Joesph was chosen at his time to do certain things as a prophet that were different from what Brigham did.
Brigham is also explaining that a person is not "ordained" to be president of the church; he is set apart to preside over the church. A person is ordained an apostle, which is a prophet, seer, and revelator. When an Apostle is chosen to be the president of the church it is by the sustaining vote of the members that acknowledges him as president, otherwise he doesn't get the job.
Just before saying this he professes his prophetic calling when he said:

"I could give you revelation about going to California, for I know the mind of the Lord upon that matter. I could give 'you revelation upon the subject of paying your Tithing and building a temple to the name of the Lord; for the light is in me. I could put these revelations as straight to the line of truth in writing as any revelation you ever read. I could write the mind of the Lord, and you could put it in your pockets. But before we desire more written revelation, let us fulfil the revelations that are already written, and which we have scarcely begun to fulfil." (JOD 6:319)

Quote #3. "It has been remarked sometimes, by certain individuals, that President Young has said in public that he was not a prophet nor the son of a prophet."(JOD 13:165)

Answer:
Here is what Wilford Woodruff said in this discourse in its entire context:

"As to President Young his labors have been with us. It has been remarked sometimes, by certain individuals, that President Young has said in public that he was not a prophet nor the son of a prophet. I have travelled with him since 1833 or the spring of 1834; I have travelled a good many thousand miles with him and have heard him preach a great many thousand sermons; but I have never heard him make that remark in my life. He is a prophet, I am a prophet, you are, and anybody is a prophet who has the testimony of Jesus Christ, for that is the spirit of prophecy."

In other words he is saying that those people were either lying or misunderstood what they heard Brigham Young say.

Quote #4. "You are now without a prophet present with you in the flesh to guide you; but you are not without apostles . . ." (Times and Seasons, 5:618)

Answer:
This was written to the church by the Twelve Apostles shortly after Joseph Smith died, so there was no one prophet leading the church at this time. Brigham Young had not yet been designated as President of the church. Here is the larger content of what was said:

"You are now without a prophet present with you in the flesh to guide you; but you are not without apostles, who hold the keys of power to seal on earth that which shall be sealed in heaven, and to preside over all the affairs of the church in all the world; being still under the direction of the same God, and being dictated by the same spirit, having the same manifestations of the Holy Ghost to dictate all the affairs of the church in all the world, to build the kingdom upon the foundation that the prophet Joseph has laid, who still holds the keys of this last dispensation, and will hold them to all eternity, as a king and priest unto the most high God, ministering in heaven, on earth, or among the spirits of the departed dead, as seemeth good to him who sent him. Let no man presume for a moment that his place will be filled by another; for, remember he stands in his own place, and always will; and the twelve apostles of this dispensation stand in their own place and always will, both in time and in eternity, to minister, preside and regulate the affairs of the whole church."

What they are saying is that they are without Joseph Smith, who is the only one who "still holds the keys of this last dispensation, and will hold them to all eternity." Joseph Smith is the only one who is the head of this dispensation and no one can take his place in that position.
A few sentences later they say that the church will have prophets:
"This church may have prophets many, and apostles many, but they are all to stand in due time in their proper organization, under the direction of those who hold the keys."