Knowledge and Belief in Medieval America
This more speculation on my part, but I've had some more thoughts on Medieval America.
The Medieval Americans, as stated before, aren't carbon copies of their historical counterparts. They're vastly different, and in many ways more advanced. Not just in things like clothing or architecture, but in health, sanitation, agriculture, and social mores.
Thanks to the fact they're the descendants of a highly advanced civilization, they will be more likely to retain some of that knowledge, even if it is common sense by our standards. For example, health and sanitation: they may not recall germ theory or bacteria, but they will at least know enough not to put your latrine next to your water source, and the importance of good waste disposal. They might even have a vestigial memory of vaccination and medicine, e.g. if you get sick with cowpox you won't contract smallpox, or if you eat this particular piece of moldy bread you'll get better.
They will also have the benefit of more advanced agriculture. They will at least have the concept of the moulderboard plow, and possible a simple mechanical reaper. More importantly, they'll have the concept of crop rotation and the importance of putting nutrients back into the soil, even if they don't know how or why it works.
Finally, they will hopefully have at least a slightly more progressive social outlook. While the feudal system is not exactly conducive to equal rights for women, for example, I can't see two or more centuries of societal progress just disappearing. At the very least, it will hopefully be less of a taboo for women to pursue things like warfare or governance.
Of course, I write this with the Feudal Core in mind. Other regions may vary. It would surprise if some were more advanced, perhaps the Buddhist or Neo-pagan areas are even more egalitarian (again, for a limited use of that word). On the other hand, some regions will almost certainly be retrogressive, like the New Israelites and the nomads.
Of course, this is not to say that the Medieval Americans will be on par with us. They will probably be very superstitious, and live in a world full of mystery and wonder. It wouldn't surprise me if some of our holidays like Halloween return to their more sinister roots. There are still ghosts and demons out there, after all.