Before cars were in invented, was "Horse-riding Under the Influence" a thing?

Yes on all three counts.

For an example, allow me to present you with this short item from the April 30, 1887 issue of the New York Times.

CHICAGO, April 29.--James Howard, a muscular young man, was sent out yesterday afternoon to break a young saddle horse for a Mr. Davis, of Wells-street. Howard was drunk, and encouraged the young animal to some marvelous feats. He made a dash up La Salle-avenue so fast "that you could hardly see anything but a streak," as the officer declares, and at the corner of Division-street urged the horse upon a passing milk wagon. There were three witnesses to swear that the horse "took" the wagon, and cleared it at one jump, but in striking the ground on the other side it dashed out its brains. Howard went headlong several feet over the asphalt pavement, and arose with no worse damage than that of a skinned nose. Justice Kersten fined him $10 for fast driving this morning, and held him pending further action by the owner of the horse.

Now, in this case, Howard was fined for "fast driving." Yes, you could be fined for speeding on a horse. City streets in the era of the horse-and-buggy could be just as crowded as they are today, and even more so in places like New York City's Lower East Side, which was notorious for its tenements.

Speeding through crowded areas could injure or kill someone, and both police and the public took a dim view of reckless behavior that endangered people. Nevertheless, the antics of drunks were just as enticing to the general public as they are today.

As the Milwaukee Wisconsin declared on Oct. 1, 1880:

"The freaks of drunken men take turns that surprise everybody, themselves included. The newspapers chronicle many of these insane freaks, but none more sensational than the performance of James Streden, an employe of the Bay View Rolling Mills."

You see, Mr. Streden managed to drive his horse and buggy onto a railroad trestle, and if you know railroad trestles, you know they don't have solid decks. They're just ribbons of iron suspended with wooden beams. They're absolutely not intended to carry a buggy with narrow wheels, nor does it seem possible for a horse to step from railroad tie to railroad tie, minding the gaps.

"The bridge has for a bottom only ties placed about 22 inches apart, and the task of walking across on a dark night like that of yesterday is quite a ticklish one, even for a sober man. For a horse, and especially for a horse controlled by a drunken driver, the task is well nigh impossible."

Needless to say, Mr. Streden's horse and buggy made it halfway across before they plunged into the river below. Man and horse swam out unharmed, but the buggy became lodged in the trestle. It was removed before the next train came by.

There's also a story from the Kansas City Times of Feb. 28, 1878, that describes how a drunken man managed to swim his horse across the Missouri River at a spot where the river was nearly half a mile across, wracked with eddies and (because of the season) full of ice chunks. Man and beast were eventually pulled up a bluff with the help of ropes.

But all this doesn't answer your question. Did people get ticketed specifically for drunken driving on a horse? Yes. Turn to the July 29, 1929 issue of the New York Times, which includes this piece:

EATONTOWN, N. J., July 28.--Arthur Berry, 53 years old, of Pine Brook was arrested today for driving a horse while intoxicated. Berry, in a light surrey, was pursuing an uncertain course up the State highway toward New York when he made a sharp left turn at Lewis Street and Johnson Avenue. The horse crashed head on into an automobile operated by Wallace Chase of Eatontown. Chase and his wife and young daughter were painfully cut when the windshield was smashed. The horse's legs were cut."

You can find even find drunken-horse-riding cases in the modern record, particularly out west. In the historical record, there's the Sept. 14, 1946 issue of the New York Times, which carries a United Press wire service account of a 56-year-old man in Detroit arrested and sentenced to 15 days in jail for "drunken driving of a horse and wagon."

If you're interested in the way horse culture was used in the United States, you might try The Horse and Buggy Age in New England by Edwin Valentine Mitchell. It was published in 1937 as a reminiscence of how things operated "when stables and blacksmiths flourished" and was targeted at an audience that grew up after the automobile replaced the horse and never experienced what life was life with the horse.