Please forgive me if I don't quite answer what you're asking. I may be misunderstanding.

The short answer to your question is "yes and yes." The LDS Church didn't suddenly come into being in a vacuum. Its converts brought what they already knew with them and Mormonism was shaped by the world it was born in, but it is a unique movement that hasn't quite followed the same trajectory as other denominations that also have roots in the 2nd Great Awakening.

They way Mormons spread their message wasn't atypical. From what I've read of early Mormon missionary diaries, they sound pretty similar to published memoirs and journals of other proselytizing Christians such as Methodist circuit riders--they established congregations in much the same manner as other denominations of the day. They held church "conferences" (I think the borrowing was originally Methodist). They preached in the manner they already knew. For instance, bystanders at a meeting in 1834 where several leading Mormons elders preached surmised from their respective styles of preaching (without knowing they were Mormon) that "they thought Joseph Young was a Methodist. . . they supposed Brigham Young was a Close communion Baptist, Orson Hyde a Campbellite or Reformed Baptist, Lyman E. Johnson a Presbyterian, Amasa Lyman a Restorationer, and Orson Pratt a Unionist" (George A. Smith, Memoirs, p. 25). So apparently they used the preaching style and rhetoric they already knew and hadn't yet developed a distinct Mormon style of preaching.

They shared doctrinal similarities with other movements and denominations of the 2nd Great Awakening too. Mormons were millenarians (similarly, the roots of modern Seventh-day Adventists is in the Millerites who believed in the 1840s that the return of Christ was imminent). They adopted a code of health that was likely influenced in part by the temperance movement. They experimented with economic collectivism and eventually reinterpreted marriage through the introduction of polygamy (similarly, both the Oneida Community and the Shakers lived communally and espoused marital relationships that were outside of the norm--respectively, complex marriage and universal celibacy).

Although Mormonism shares a lot with other denominations of the time period, it is unique. I can't recall another church during the 2nd Great Awakening that produced a body of scripture as large and complex as they did. Some particular doctrines are unique to Mormonism too, such as the 3 degrees of glory in heaven (see Doctrine and Covenants, sec. 76), and I can't think of another group that took the notion of gathering god's people together to build the city of Zion as literally as Mormons did (as evidenced by constant proselytizing efforts and successive attempts over 15 or so years to build cities in Kirtland, Ohio; Independence and Far West, Missouri; Nauvoo, Illinois, and eventually Salt Lake City, Utah).

A classic book on the Second Great Awakening is Nathan O. Hatch's The Democratization of American Christianity (New Haven, CT: Yale University Press, 1989). As I recall, he asserts that Mormonism was a populist movement and emphasizes the economic side of it.

Another book that may be of interest is Richard Lyman Bushman's Joseph Smith: Rough Stone Rolling, A Cultural Biography of Mormonism's Founder (New York: Alfred A. Knopf, 2005). It's a bit of a brick (over 500 pages), but I think it does a good job putting Joseph Smith into cultural context. Being a biography, the focus is squarely on Smith and the LDS Church, but any good biography will give good insight into the subject's worldview and the things that influenced them.

Origins of mormonism aren't often discussed by modern members, because many of the actual events tarnish the image of their church. Most members aren't aware that the original prophet Joseph Smith transcribed the book of mormon with a "seer stone" which he placed into a hat. This is the same stone that JS used to "find" water and gold at the time.

Prior to establishing mormonism, JS was known to the state of New York as "Joseph the Glass looker" and had been charged with fraud and being a "disorderly person."

The burned over district gave rise to several religions, including mormonism. It's difficult to find a lot of neutral media concerning mormonism. Media created on the topic is almost always slanted either for or against mormonism.

Early mormonism appealed strongly to rich white men. Essentially it started as a sex cult in which men would marry multiple (and often adolescent) wives. This attracted enough people to form an organization, but was obscure enough that they were driven west.

PBS did one of the best documentaries I have seen on the topic, though it is soft on criticism. See it here.

Edit: looks like a few TBMs have been here.