Well, they actually had horses long before that. But the natives ate them into extinction rather than riding them.
Horse ancestors originated in NA. But disappeared long before any Indians or other humans were here.
Not likely, they died with all the other megafauna during the beginning of the Younger Dryas.
There’s a range of opinion of course, of course.
There’s a very small amount of scientifically dated remains of humans and animals from North American (and S A and the Caribbean for that matter) that antedate 1492.
Artifacts and petroglyphs related to horses have mostly not been scientifically dated (and thanks to NAGPRA etc, probably never will be), and oral traditions are, uh, fluid and not generally trustworthy, irrespective of their geographical and ethnic source.
Anyway, here’s one take on the “we had horses before Columbus” view:
https://ictnews.org/news/yes-world-there-were-horses-in-native-culture-before-the-settlers-came
They appeared back on it when the Spanish Conquistadors arrived here with them.
Up until that time Indians had never seen one and were scared to death of them at first.