Members of the Church of Jesus Christ of Latter Day Saints are encouraged to get a testimony of each of those prophets.
1.I don't disagree with you. However, I would think that a testimony of the current prophet, would be enough. If you know the first to be true, and today's to be true that should be enough. If you decide that one in the middle isn't a true prophet, but you have a testimony of the current one, does that mean the entire church is now wrong but led by a true prophet? No, that wouldn't make sense. Personally, I would Look at Joseph Smith Jr. and the current prophet before joining, and look at the others after joining. If you change your mind latter, you could always leave.
It boils down to a question of what do you think is the bigger sin, ignoring a prompting from the holy spirit, or following a prompting before studying it out? and I don't think I can answer that here.
EDIT So I was pointed to a Statement by a previous president, I would think this is a shortcut, gain a testimony of this and they you can not worry about the other prophets.
“The Lord will never permit me or any other man who stands as
President of this Church to lead you astray. It is not in the
programme. It is not in the mind of God. If I were to attempt that,
the Lord would remove me out of my place” (Teachings of Presidents of
the Church: Wilford Woodruff , 199).
2.They do encourage this from what I understand. There is even a manual about this. It is an "institute" manual which I believe means it is a college level course. The introduction states:
The purpose of the Teachings of the Living Prophets Student Manual is
to help you strengthen your testimony of living prophets and their
teachings. It discusses our need for living prophets, the role of the
President of the Church, the divine order of succession in the
Presidency, the Quorum of the First Presidency and the Quorum of the
Twelve Apostles, and the importance of general conferences. President
Harold B. Lee (1899–1973) taught, “It is not alone sufficient for us
as Latter-day Saints to follow our leaders and to accept their
counsel, but we have the greater obligation to gain for ourselves the
unshakable testimony of the divine appointment of these men and the
witness that what they have told us is the will of our Heavenly
Father” (in Conference Report, Oct. 1950, 130). A witness of the
divine appointment of the living prophets is promised to each of us as
we study the counsel of the Brethren and resolve to sustain them
through our obedience. This manual will assist you in your study of
the living prophets.
They also have a manual with the teachings from each prophet and it is another institute manual. Meaning, I think, They study all of them. They also have a larger book on each prophet (I think it's all of them), Scroll to the bottom of this page for the list of links.
3.From what I understand, they just ask you to pray about it and ask God. Seems pretty simple and straight forward. I would guess, that if the holy spirit already helped you out with a testimony of the current prophet, Joseph Smith Jr. and the Book of Mormon, this wouldn't be much more to ask.
EDIT After talking to a few more members I know, I was pointed to a "seminary" manual, which I believe is used with high school students. Page 158 of this manual. I would recommend you read the entire page about succession. The part below talks about the first conference that Brigham Young held:
“Brigham Young then spoke briefly, comforting the hearts of the
Saints. As Brigham spoke, George Q. Cannon remembered, ‘it was the
voice of Joseph himself,’ and ‘it seemed in the eyes of the people as
if it were the very person of Joseph which stood before them.’ William
C. Staines testified that Brigham Young spoke like the voice of the
Prophet Joseph. ‘I thought it was he,’ Staines said, ‘and so did
thousands who heard it.’ Wilford Woodruff also recalled that wonderful
moment and wrote, ‘If I had not seen him with my own eyes, there is no
one that could have convinced me that it was not Joseph Smith, and
anyone can testify to this who was acquainted with these two men.’
[Quotations in History of the Church 7:236.] This miraculous
manifestation, seen by many, made clear to the Saints that the Lord
had chosen Brigham Young to succeed Joseph Smith as leader of the