You call them by their nationality. There used to be different names based on percentage of mixing, but today that is considered to antiquated and obsolete. People now use the umbrella term...
They didn't treat them for anything. If they had, you would be living in a well-populated society (possibly in your home country, or maybe here) within a vibrant, healthy, and modern Native culture....
Until the early 1500's there were no horses in North America. The ones that were lost or escaped during Spanish campaigns became the wild horses that Amerindians began to use over the next 400 years.