If you think of it, the Mississippi basin is practically China. Two vast rivers and many smaller yet still grand rivers make for perfect transport networks and could allow for an empire from between the Rockies, Appalachians, Great Lakes, and Rio Grande. The north is too cold for agriculture and the residents savages not worth conquering--perhaps like China and Siberia. The east coast is too remote to exert much control, but will be culturally similar--like China and Korea/Japan. The west is too dry for much agriculture and the soil rough and rivers seasonal. It's like Mongolia or Central Asia. The south is also too dry and too rugged, and part of a different cultural area--like Tibet and India.
With a POD far enough in the past, you could probably assemble this great empire. I'd go with an Eastern Agricultural Complex wank, although it might take surviving American horses since deer probably aren't a substitute and elk, moose, and reindeer can't survive outside of the more northern parts of this range, while bison can't survive in captivity without modern medicine.
What I could see is that within this region, you might have a division comparable to North and South China. The north is somewhat agriculturally poorer, but much wealthier in resources, with plenty of copper and iron. The south is rich in agricultural potential, but has a relative lack of iron (although if/when coal comes into prominence, things will change). The south also has great trade potential.
So the north conquers the south, and over the centuries assimilates the ethnic groups there into the culture and language of the northerners. Gradually the economic and political center shifts south thanks to demographics and trade with Mesoamerica.
Best capital would probably be something centrally located, not on the fringes, while also at a good river crossing. While there will be a ton of capitals (look how many China went through), I'd pick a site at St. Louis first as my "Beijing", assuming the Plains aren't full of horse nomads. If they were, then St. Louis isn't as good (but still very nice) and Muscle Shoals, AL might be nicer due to more protection. Any good site on the Tennessee, Cumberland, or Lower Ohio, provided flood protection doesn't need too much investment and it controls a good crossing, would be a great capital in this case.