The Americas contain at leas two centers of agriculture: Mesoamerica and the Andes/Peruvian Coast (and Eastern Agricultural Complex as noted by /u/Reedstilt). I'm more familiar with the former, and it is where maize was domesticated, so I'll focus there.

Maize is a domesticated form of teosinte, a grass indigenous to Mesoamerica. Like with all grain domestication events, we see a selected evolutionary process which increased kernel size, nutritive value, and retention. The last point is a bit vague, but the important thing to remember is that kernels on domestic plants tend to "cling" tightly to the rachis. A common feature to domestication of maize, wheat, rice, and all other grain crops is the plants were gradually selected to produce kernels which less easily "shattered," separated from the rachis, and therefore preserved the nutritious kernels for human harvest.

Regarding the increase in kernel size (and cob size, specific to maize), we being to run into the problem of when. Evolution works on gradients, not discrete categories, which makes the distinction between ancestral and modern forms an eternal grey area. UGA has a page which has some comparison photos of modern maize and extant teosinte plants, both of which they categorize as genetic subspecies of Zea mays. The question of when the cobs grew large enough and the kernels numerous to be called "maize" and not "teosinte" is one that will trouble archaeologists and archaeobotanists for many years to come, particularly since our earliest evidence is reliant less upon preserved specimens and more on palynological (pollen) samples and phytoliths.

Moving to the archaeological evidence, we have fairly definite evidence that domestication of a plant we could call maize was already underway some 5-6K years ago in present day Oaxaca, Mexico. Benz (2001, see also Benz & Long 2000) presents actual fossilized cobs. They are nowhere near the size of modern day cobs, but they are recognizably distinct from wild teosinte and similar to modern day maize. Of course, Piperno et al. (2009, see also Ranere et al. 2009, on which Piperno was second author) have found microscopic phytolith evidence that domestication was already proceeding in the Balsas River valley in the neighboring state of Guerrero almost 9K years ago.

The thousands of years between those two dates may seem like a problem, but they are really not. Again, when we look at any domestication event, we very rarely see sharp delineations. Proposing that people where utilizing a maize progenitor 9K years ago, then were using something definitely maize-like 6K years ago, is not only not contradictory, it is expected. Maize was still being selected for more nutritious and efficient traits for centuries, even as the Olmecs were arising as Mesoamericas first complex society. Arnold (2009) has even posited that it wasn't until around 900 BCE that maize really achieved near modern levels of productivity, and that this was part of the reason the Olmec cultural center shifted from San Lorenzo to more maize-amenable sites like La Venta and Tres Zapotes. The time between domestication and complex societies is really a question of adoption as a staple crop, and the relatively late adoption of maize as a staple crop is a prevailing theory in Mesoamerican studies, suggesting instead that lower yield maize was used as prestige crop, perhaps as a fermented chicha-like beverage (Webster 2011 is a good summary).

This is just one crop. Admittedly, a very important crop, but we can find similarly deep and interesting histories for tomatoes, beans, squash (probably domesticated even before maize), cacao, amaranth, chayote, chia, and a long list of other plants. This also does not touch on innovations like chinampas or events like the spread of maize to South America or into other parts of North America, where it displaced autochthonous agricultural crops.

Yes and no, it all depends on which group of people you are looking at because not everyone practiced agriculture and not everyone practiced agriculture all the time.

A great overview of the major foods available to the Aztec, Maya, and Inca can be read in Sophie Coe's book America's First Cuisines from 1994. Coe notes that there are many foods she had missed like herbs and in the last 20 years archaeology has added to the list. It is a great start to exploring the subject further if you are interested.