Buttons on the coat sleeve are a carry-over from military uniform styling. Three buttons on the seam line common.
They started out functional on a uniform or sometimes denoted rank, but in transition to civilian styles became decorative.
(As far as riding splits, I bet since man has domesticated the horse his apparel has been modified to sit a horse.There were riding splits on tunics, mail, etc. since earliest times.)
Think of our Revolutionary era where long coats were split into tails with turnbacks of contrasting color, in uniforms it was often a unit identifier along with facing color on lapels.
Formal wear today even has swallow-tail coats. As prevailing style shortened the skirt on coats, they still kept the idea of a split. Sure makes sitting in a chair more comfortable, too.
Hard dates and further data will be contributed soon, I'm sure. But those details of style go way back before our era that we try to depict.