To be fair, a horse and buggy is somewhat self-driving. The horse knows how to walk and stop and when to make a noise if something goes awry.

Can the horse and buggy automatically move out of the way if someone else is about to hit it?

Yes. It is a horse. Horses are completely capable of independent movement.

Yes but you shouldn't rely on it and should always be paying attention. We're hoping that the next version of horse should be fully autonomous.

Within the limitations of the horse's reaction time and maneuverability with an attached buggy, yes.

And those limitations are severe, given the speed limits of many rural roads. Even at 35 MPH, a car is 3x as fast as a horse and buggy.