Do LDS believe in Original Sin after all?

To understand the differences here between mainstream Christianity and LDS doctrine on original sin and how this impacts humanity, it is first necessary to grasp that they believe Adam had a pre-mortal existence. This first quote from their own sources shows that (basically) Adam did humanity a favour by 'transgressing' - and note how your own quote has the LDS saying he transgressed, but that humans sin personally. They don't say Adam sinned - he transgressed. We sin. This subtle but deliberate use of different words is important, as these quotes show:

“…Adam and Eve were taught the gospel long before they were in Eden. Indeed, life before Eden was obviously a time of preparation for their important ministry. In the premortal realm, Adam was known as Michael…

The implications of the Fall are often misunderstood in the contemporary world. One misperception the world has about the transgression of Adam and Eve is the concept of original sin…

Our Father in Heaven knew Adam and Eve would fall. In fact, the Prophet Joseph Smith taught that Adam “was made to open the way of the world” (Teachings, 12). Lehi tells us, “Adam fell that man might be; and men are, that they might have joy” (2 Nephi 2:25). Modern scripture makes it clear that it was the will of the Father, as part of his plan, that Adam and Eve transgress and thus be moved out of Eden…

By wearing their coats of skins, they would be reminded that the Atonement covered their sins.” https://www.lds.org/study/ensign/1998/01/what-modern-revelation-teaches-about-adam?lang=eng [See the summary list at the end of the article.]

This second quote is also about pre-mortal existence and that Adam's 'transgression' was necessary, but that it wasn't a sin :

"We know that there was a council of the Gods in which the plan of our Eternal Father was sustained... This plan presupposed that Adam and Eve would fall from the Garden of Eden, so it provided for the Savior (see Alma 34:9-10, 14-15), a mediator who would provide the means whereby we could succeed in this earth-life experience and return to our Father in Heaven prepared for the next phase of our development (see Alma 12:24).

President Joseph Fielding Smith wrote: “It is true that Adam helped to form this earth. He labored with our Savior Jesus Christ. I have a strong view or conviction that there were also others who assisted them. Perhaps Noah and Enoch; and why not Joseph Smith, and those who were appointed to be rulers before the earth was formed?...

Some even erroneously think that Adam and Eve’s transgression was sexual in nature. They assume that none of the general conditions we find on earth now would ever have come if our first parents had not been sinful… In contrast to most readers of the Bible, we believe that Adam and Eve both should be commended for what they did to bring about the Fall. We understand that without the Fall none of us could have come to the earth and the whole plan of salvation would have been frustrated (see 2 Nephi 2:25). …

The Prophet Joseph Smith referred to their choice to eat of the fruit as a “transgression,” not a sin (Articles of Faith 1:2). Similarly, Elder Bruce R. McConkie of the Quorum of the Twelve Apostles explained: “It is proper and according to the scriptural pattern to speak of the transgression of Adam, but not the sin of Adam. (D. & C. 20:20; 29:40 Second Article of Faith)… Knowledge of good and evil is an essential element in the commission of sin, and our first parents did not have this knowledge until after they had partaken of the fruit of the tree of knowledge of good and evil. Thus, by being required to leave the Garden of Eden, Adam and Eve caused the great plan of happiness to go forward. Mortality came to all living things; procreation began the process of bringing us, the sons and daughters of God, to earth as Adam and Eve’s posterity.” https://www.lds.org/study/ensign/1998/01/in-the-beginning-a-latter-day-perspective?lang=eng

These quotes show why the LDS can claim that 'the Fall' was a positive thing, and why the LDS teaches that Adam did not 'sin in the flesh', although he 'transgressed' - but necessarily so, without which the whole plan of salvation would have been frustrated. This sentence from the quote above states it succinctly: "...we believe that Adam and Eve both should be commended for what they did to bring about the Fall. We understand that without the Fall none of us could have come to the earth and the whole plan of salvation would have been frustrated."

If Adam is to be commended for "the Fall", and if Adam did not sin, then this turns the whole mainstream doctrine of "Original Sin and The Fall" on its head. It is no wonder that others are confused by LDS statements compared with what their Book of Mormon says, and have to ask for clarification.

It would be helpful if LDS believers could explain the quote you gave (Moroni 8:8) in light of that verse in the quote above (2 Nephi 2:25), "Adam fell that men might be; and men are, that they might have joy." Also, how this verse in 'Pearl of Great Price', Moses 5:10,11 sits with the verse you quoted: "Adam cried, 'Because of my transgression my eyes are opened and in this life I shall have joy'." They only show why that Article of Faith 1.2 speaks of Adam transgressing, but not sinning.

Partially, the doctrine is unique from other denominations. The Church of Jesus Christ believe:

• Adam sinned/transgressed, bringing about the Fall (mankind separated from God's presence-became mortal and subject to sin/death¹)
• Mankind is now in a fallen state because of the Fall (separated from God's presence-and accountable for own actions)
  • Children's sins/transgressions are covered by the Atonement-if they even have the knowledge (not accountable-²)
• Rest of us though, are not accountable for Adam's mistake, just our own³

_{1 Guide to Scriptures: Fall of Adam and Eve, Alma 12:12, The Fall and Infinite Atonement, Bible Dictionary: Fall of Adam and Eve}

_{2 Salvation of Little Children, Elder Bruce R. McConkie 1978, Moroni 8:8,10,20}

_{3 Gospel Topics: Origin Sin}

Preparing this answer was my first time looking up how "original sin" is formally defined, and if I understand right, its definition may not be consistent across Christianity. So depending on your definition, the answer varies.

Do Latter-day Saints believe that we inherit some kind of intrinsic sin or guilt at birth as a result of the Fall of Adam?

No. You quote the second article of faith,

We believe that men will be punished for their own sins, and not for Adam’s transgression.

Additionally, from Moses 6:53-54,

53 ...And the Lord said unto Adam: Behold I have forgiven thee thy transgression in the Garden of Eden.

54 Hence came the saying abroad among the people, that the Son of God hath atoned for original guilt, wherein the sins of the parents cannot be answered upon the heads of the children, for they are whole from the foundation of the world.

Do Latter-day Saints believe that we inherit some kind of sinful state, or proclivity to sin, as a result of the Fall of Adam?

This one is also a no. From 2 Nephi 2:25-27,

25 Adam fell that men might be; and men are, that they might have joy.

26 And the Messiah cometh in the fulness of time, that he may redeem the children of men from the fall. And because that they are redeemed from the fall they have become free forever, knowing good from evil; to act for themselves and not to be acted upon, save it be by the punishment of the law at the great and last day, according to the commandments which God hath given.

27 Wherefore, men are free according to the flesh; and all things are given them which are expedient unto man. And they are free to choose liberty and eternal life, through the great Mediator of all men, or to choose captivity and death, according to the captivity and power of the devil; for he seeketh that all men might be miserable like unto himself.

The redemption of Christ has made us free to choose whether to follow Him or to follow the devil. Now, this doesn't mean that we don't have weakness or other personal defects that lead us to sin, but I would argue from the scripture above that these of themselves are not effects of the Fall of Adam. Others may have additional perspective to give on this point, and I'd be happy to hear it and update my answer. There are certainly scriptures that talk about man becoming carnal/sinful by nature, but I'd still argue from the scripture above that those aren't direct effects of the Fall of Adam, at least to the degree that we avail ourselves of the redemption of Jesus Christ.

Do Latter-day Saints believe that there are negative effects of the Fall of Adam that, without the redemption of Christ, we would inherit by being born into this world?

Yes. The Fall of Adam brought about the separation of Adam and Eve and children from God. From Alma 42, various verses between 7 and 15,

7 And now, ye see by this that our first parents were cut off both temporally and spiritually from the presence of the Lord; and thus we see they became subjects to follow after their own will.

9 Therefore, as the soul could never die, and the fall had brought upon all mankind a spiritual death as well as a temporal, that is, they were cut off from the presence of the Lord, it was expedient that mankind should be reclaimed from this spiritual death.

14 And thus we see that all mankind were fallen, and they were in the grasp of justice; yea, the justice of God, which consigned them forever to be cut off from his presence.

15 And now, the plan of mercy could not be brought about except an atonement should be made; therefore God himself atoneth for the sins of the world, to bring about the plan of mercy, to appease the demands of justice, that God might be a perfect, just God, and a merciful God also.

The "temporal death" here is the separation of a person's spirit from their body, and "spiritual death" is the separation of a person from God. Both of these needed to be overcome for people to be able to return and dwell in the presence of God once again. Jesus Christ has provided redemption from temporal death free to all people without condition by one day resurrecting them all. Jesus Christ has provided redemption from spiritual death conditioned on obedience to the laws and ordinances of His gospel. But without Christ and the redemption He has made, mankind would forever be separated from God as a result of the Fall of Adam.

To return to Moroni 8:8 which is the basis of your question,

Listen to the words of Christ, your Redeemer, your Lord and your God. Behold, I came into the world not to call the righteous but sinners to repentance; the whole need no physician, but they that are sick; wherefore, little children are whole, for they are not capable of committing sin; wherefore the curse of Adam is taken from them in me, that it hath no power over them; and the law of circumcision is done away in me.

The curse of Adam is this separation from God, caused by a combination of both temporal (or physical) death and spiritual death. This definition is, I think, meant to be understood in context of other scriptures, particularly from the Book of Mormon and some of which I shared in my answer. At the very least, this teacher's manual for Moroni 8 says as much.

I would add as well, that in Moroni 8:8, the speaker (Mormon) indicates that this "wholeness" of little children is with them from birth. So even if the "curse of Adam" meant something akin to original sin, it couldn't even be that because the scripture indicates that this curse is taken away from little kids, and they are whole in Christ. This is the state they are born into, without any prerequisite (such as baptism), and they do not inherit any curse from Adam.

A good summary, as linked by @depperm, is the short Gospel Topics entry on original sin.