Yes. They've also caused three major epidemics in the past two decades - there's no reason they shouldn't have emerged historically from time to time.

So, a likely scenario is: a new virus appeared, it swept through the population, caused serious disease, a bunch of people died, the survivors had better resistance to it, it settled as a mostly innocuous thing (such as the common cold) or disappeared completely, and the next generations remembered it as "the plague of emperor Fancy Pants" or something.

Yes, people seem to miss this. There are several mild endemic coronavirus diseases in humans, usually called "common cold" along with many others.

Yes. But then again they didn’t start out as endemic virus but as a pandemic one. They probably wrecked a lot of havoc before becoming just a cold. Mayas and Aztecs can tell.