True, but trump driving a car would be. Or so spake SCOTUS. Once l’etat cest moi, everything is an official act; yes, even pooping.
Perhaps not. I could certainly imagine some sort of law about reckless horsemanship in a crowded city. No galloping where pedestrians are about, etc.
Back in the days of carriages with 2 or more horses, carriage collisions and run-over pedestrians were not rare occurrences. IIRC from my late wife, a lot of the common-law jurisprudence about motor vehicle accidents dates from the horse-drawn days. If there’s civil law about it, good bet there was code law too.