That doesn't answer the stripe question....if all horses evolved, then the stripe gene came in when??...it would seem to be at a specific time.
I was just answering your question where they fit in the equine family. Go use google yourself.
A successful preditor focuses on one individual ("cutting one from the herd") because that one individual will eventually wear out.
The stripes are an adaptation used (by the herd collectively) because it makes a particular animal hard to follow by a given preditor, making this "cutting" process difficult.
I could see how such a pattern would easily evolve, in that those without such a pattern would get eaten and not have babies.