Popular history states that the myth of the centaur arose when early Greeks first encountered warriors on horseback. Did any similar legends arise after the first Native American contact with European cavalry? How was the arrival of horses represented in Native American legends?

I disagree with the premise. Although I have also seen the assumption put forward that the centaur was inspired by early Greeks encountering people on horseback, I have never seen that idea proven. It really amounts to nothing more than speculation.

Europeans had a widespread tradition about the ability of people to turn into animals, animals to turn into people, and of the existence of hybrids - combinations of animal and people. We would not suggest that beliefs in merfolk were inspired by the first encounter of people swimming in the ocean. Similarly, the many variations of human-bodied, animal-headed entities in Egyptian art was not the result of various combinations or misconceptions in reality.

It is a common belief (with no real basis in reality) that "all legends are grounded on something." This belief, which often manifests in this sub, has inspired all sorts of searches for the "truth behind the legend." Belief systems do not require something real upon which to grow. Myths and legends more often that not simply "are," without any hidden basis lurking in the distant past.

That said, I'll leave it to someone else to address Native American oral tradition.