@JavelinJoe@Khoopa Generally you don't let your crotch touch the actual rod period. When (for example) going over a fence, you squeeze your thighs in a certain way to prevent that. Takes more energy but the protection is needed. Or you just wear cup in the worst case.
Side saddling tends to be used in cases where you're expecting to constantly get on and off of a horse, or just for fashion. Obviously it's a lot less stable.
No, women traditionally rode side-saddle because they were expected to wear long skirts. Trying to sit on a horse/broomstick/etc normally while wearing a skirt would cause it to get hiked up above the knee, which was a big taboo back when people actually rode horses out of necessity. That's also why female bicycles have the drooping top bar, because it accommodates wearing a skirt.
Any contemporary woman riding something while wearing pants would sit normally. It has nothing to do with genital configuration, and their clit wouldn't rub on anything anyway, unless they were intentionally leaning far forward.