What has the LDS Church (Mormons) historically taught about the Curse of Cain and dark skin?

The Mormon "Book of Moses" 7:22 states that (emphasis added):

And Enoch also beheld the residue of the people which were the sons of Adam; and they were a mixture of all the seed of Adam save it was the seed of Cain, for the seed of Cain were black, and had not place among them.

Mormon viewpoint for most of their history was that black people were banned from priesthood and from some other blessings that white people were entitled to. Because of this curse and statements elsewhere in the Book of Mormon (such as 2 Nephi 5:21-23) if a white man married a black woman their children would not qualify for priesthood.

In fact the Utah Territory Slave Code of 1852, signed into law by Brigham Young (President of the LDS Church from 1847 until his death in 1877; Governor of the Utah Territory from 1851 to 1858), said the following about sexual intercourse between a white person and an African person (sec. 4, available here):

and if any white person shall be guilty of sexual intercourse with any of the African race, they shall be subject, on conviction thereof to a fine of not exceeding one thousand dollars nor less than five hundred, to the use of the Territory, and imprisonment, not exceeding three years

(According to Wikipedia, the slavery portion of the act was repealed by Federal law on June 19, 1862 but the portion quoted above remained in effect until 1963.)

It is noteworthy that Brigham Young was the Prophet, Seer and Revelator of the LDS Church during his time as Church President. He is for this reason viewed with reverence by the faithful even today and his opinions on "Negroes" have an effect on the Mormon view of their past race policy.

For example, Brigham Young wrote in a discourse titled "We Must Believe in Slavery" on 23 January 1852 (available e.g. here) as follows (second to the last paragraph):

When a master has a Negro and uses him well, he is much better off than if he was free. As for masters knocking them down and whipping them and breaking the limbs of their servants, I have as little opinion of that as any person can have; but good wholesome servitude, I know there is nothing better than that.

Regarding interracial relationships, Brigham Young wrote in the Journal of Discourses, vol. 10, p. 110 in 1863 that:

"Shall I tell you the law of God in regard to the African race? If the white man who belongs to the chosen seed mixes his blood with the seed of Cain, the penalty, under the law of God, is death on the spot. This will always be so."

Closer to our modern era it seems that the Mormon church had implemented a "one drop rule" to determine whether a male was qualified for the priesthood. Rather than having a rule that a drop of white ancestry makes the individual qualified for the priesthood, the policy was that a drop of "negro" ancestry makes the individual disqualified.

Regarding this the Mormon Apostle Mark E Petersen gave a lecture on on August 27th, 1954, titled "Race Problems As They Affect The Church" in which he mainly spoke about the LDS Church relationship to what at that time was referred to as "Negroes".

The text of above lecture can be found here at archive.org. The page references for the quotations from the lecture refer to this document.

Regarding segregation Petersen stated in this lecture (pp.14-15) that

The Lord segregated the people both as to blood and place of residence, at least in the bases of the Lamanites and the Negroes we have the definite word of the Lord himself that He placed a dark skin upon them as a curse -- as a sign to all others. He forbade intermarriage with them under threat of extension of the curse (2 Nephi 5:21) And he certainly segregated the descendants of Cain when He cursed the Negro as to the Priesthood, and drew an absolute line. You may even say He dropped an iron curtain there. The Negro was cursed as to the Priesthood, and therefore, was cursed as to the blessings of the Priesthood. Certainly God made a segregation there.

On the curse of Cain he quoted President Wilford Woodruff who served as the 4th president of the LDS church as follows (p.18):

President Woodruff added, "The Lord said, 'I will not kill Cain, but I will put a mark upon him, and that mark will be seen upon every face of every Negro upon the face of the earth.' And it is the decree of God that that mark shall remain upon the seed of Cain, until the seed of Abel shall be redeemed, and Cain shall not receive his priesthood until the time of that redemption. Any man having one drop of the blood of Cain in him cannot receive the Priesthood."

From reading this lecture it seems that segregation of the races in USA was viewed as a protection for the church as it would reduce the risks of inter-marriage. At the end of the lecture (last page of the document), the speaker said

There are 50 million Negroes in the United States. If they were to achieve complete absorption with the white race, think what that would do. With 50 million Negroes inter-married with us, where would the priesthood be? Who could hold it, in all America? This what that would do to the work of the Church!

This same sentiment can be seen in a letter, dated April 16, 1964, from the First Presidency to a Bishop in Idaho (the letter is available here).

The letter addresses the concern that a sister who had two boys approaching priesthood age may have had some "negro blood":

We have read with interest your letter of April 6th in which you refer to a certain family in your ward about whom there has been some discussion as to whether or not they have negro blood. You indicate that there are two boys who are approaching priesthood age and that it has been rumored that the mother has negro blood, although there are few, if any, visible signs thereof. You also tell us that according to this sister's statement her case has been investigated on previous occasions and no evidence has to date been disclosed that would indicate that she has negro blood. We note your quotation from a patriarchal blessing given to this sister, [name withheld], in which the patriarch states that she is "of the daughters of Israel through the loins of Ephraim," and declares this to be her lineage.

Inasmuch as there is no conclusive evidence that these people have negro blood it would seem that you would not be justified in denying the boys the privilege of being ordained in the priesthood in the event they are otherwise worthy. We do not feel that the burden of proof rests upon this sister and her family.

It seems that if there had been "conclusive" evidence that the mother had "negro blood", her boys would have been denied the priesthood rights on that basis.

The document "THE RECEIPT OF THE REVELATION OFFERING THE PRIESTHOOD TO MEN OF ALL RACES AND COLORS" by Bruce R. McConkie (available here), dated June 30th, 1978, discusses (in par. 1) the problems this policy led to when the LDS Church opened a temple in Brazil:

There have been discussions for instance, about patriarchal blessings for Negroes and about the problems we might encounter in Brazil after the dedication of the Sao Paulo Temple. As a matter of historical fact, many of the early settlers in Brazil were Portuguese and African Negroes. There was extensive intermarriage. It is generally felt that an appreciable part of the Brazilian nation is now composed of people who have some degree or other of Negro blood in their veins. Many of these have no facial or other physical characteristics to indicate in any way that they have Negro ancestry. Many of our Church congregations, particularly in Northern Brazil, have in them full and part blood Negroes. The social problems existing in the United States are not found in Brazil to any appreciable degree and so this intermixture of races has not caused problems except with reference to priesthood ordinations and the prospects of temple blessings.

In light of above, it seems that both the expansion in Brazil and the public opinion against the church here in USA were factors that led the policy to be abandoned.

The problem for the Mormon church however is that it is difficult for them to explain the reasons for the policy without making either their past leadership or their God look racist.

About this, The New York Times article "Why Race Is Still a Problem for Mormons", dated August 18, 2012, states (in par. 10) that

while Mormon leaders generically criticize past and present racism, they carefully avoid any specific criticism of past presidents and apostles, careful not to disrupt traditional reverence for the church’s prophets.

Consequently it looks like the approach that was adopted was to not to try to explain the past stance at all.

This may be why the Mormon Elder Holland said, when asked in an interview with PBS about the "folklore" banning blacks from the priesthood, as follows (emphasis added):

One clear-cut position is that the folklore must never be perpetuated. ... I have to concede to my earlier colleagues. ... They, I'm sure, in their own way, were doing the best they knew to give shape to [the policy], to give context for it, to give even history to it. All I can say is however well intended the explanations were, I think almost all of them were inadequate and/or wrong. ...

It probably would have been advantageous to say nothing, to say we just don't know, and, [as] with many religious matters, whatever was being done was done on the basis of faith at that time. But some explanations were given and had been given for a lot of years. ... At the very least, there should be no effort to perpetuate those efforts to explain why that doctrine existed. I think, to the extent that I know anything about it, as one of the newer and younger ones to come along, ... we simply do not know why that practice, that policy, that doctrine was in place.

Update: there was also this "A Proclamation To The World" about "the negro", dated August 17, 1949 (also here at archive.org).