Many of the great rivers of Asia (Indus, Ganges, Brahmaputra, Mekong, Yangtze, etc) developed enormous populations dependent on them. In fact some of these rivers were cradles of civilization. Yet it seems like the Mississippi River was never exploited in this way by Native Americans. Why was this difference?
http://en.wikipedia.org/wiki/Mississippian_culture
http://en.wikipedia.org/wiki/Cahokia
The records of Hernando de Soto's expedition to what is now the american south paint a vivid picture of settled towns with wooden walls, agriculture, and societal development in the valley region. While it wasn't as population dense as the regions you mentioned in Asia, it does raise the question of what could have happened had the evolution of native societies been allowed to continue, and not been cut short by Old World diseases and other biological invaders.
Of note is the fact that animal and plant domestication arrived relatively later to the Americas (and the Mississipi River region in particular) than it did in China (or the rest of Eurasia). The plants and animals that were used simply weren't as energy-dense as in the asian ones you mention. Modern society is totally dependent on surplus calories and being able to free up part of your population from producing food, and devote them to other endeavors such as construction, war, technological development, and so on. Without the agricultural foundation upon which to build, these others simply will not follow, or at least not flourish to the same degree. This, as much as anything, is the answer to your top question.
There were many civilizations that utilized the Mississippi River as a source of growth, like many of the other civilizations across the world used rivers as an agricultural base for growth. We are finding more and more about this through archaeology, but I'll focus on one civilization in particular: Cahokia.
Cahokia was the center of Mound Building Mississippian civilization in North America. From what we can understand about it through archeology, they had a regimented class system, with elites that lived on the massive mound (40 football fields or so in length and width) and "subdivisions" of different communities around the mound. Some population estimates put the maximum population as high as 1 million souls, with a complex culture, theology, and tool making capacity.
However, this civilization did not survive, like almost all of the major civilizations of North America, due to virgin soil epidemics of smallpox, and a variety of other diseases hitting at the same time. Some estimates place the death rate total for all of these diseases hitting at once at 95%. With this much of the population dying, these civilizations fell apart, and the produced upon land overgrew with forest and overgrowth. The survivors of these epidemics came together in tribes, and became the Native American tribes that future European and American colonists would interact with. Europeans would see the "unused" land and think them uncivilized.
http://en.m.wikipedia.org/wiki/Cahokia
McNeill, William Hardy. Plagues and Peoples. Garden City, NY: Anchor, 1976.
I have more sources from a course pack from a class I took on Native American history to 1840. Unfortunately I do not have it with me, but I could find those sources for any that are interested.
Like r/cbd03b said, there was a culture that lived and used the Mississippi River -- the Mississippian culture, also known as the Mound Builders. They thrived from around 1000 - 1540 CE and had vast populations, but had mostly collapses/dispersed by the time European colonialists reached the area. There has been evidence that they participated in a vast trading system with the Ancestral Pueblo/Anasazi in the Southwest and the Maya in Mesoamerica.
It's actually interesting to note that one of the reasons you may have been unfamiliar with the Mound Builders is because of the politics of early archaeology and manifest destiny. Even though early archaeologists in the 1800s began to find evidence of the Mississippian cultures (I mean some of the mounds are huge and pretty hard to miss as man-made, they frequently ignored the evidence or looking for alternative explanations that didn't have to do with a well-established and sophisticated culture related to the Native Americans. This was because of manifest destiny, which was the principle that if the land wasn't been used well by the Native Americans, the Europeans had the right to take and use it themselves (and obviously I'm simplifying). Finding evidence of the Mississippian cultures would disprove the idea that the Natives did not have the capacity to hold a civilization and had no right to the lands in America. Take a look at "alternative explanations" on this wiki page. I mean, even calling their structures "mounds" instead of "truncated pyramids", which is what they are is because no one wanted to admit there were pyramids in the lands they were taking over. Even today, the Mississippian Cultures are relatively unknown compared to the other cultures and civilizations you mentioned.
