] How common was it for people to ride horses during the Regency period? Were they only for the gentry and aristocracy?

Generally yes, because they were the only ones who could afford them. This can be seen, for example, in how wealthier people had more carriages etc, becaus ethey had more horses to be used. the poorer people, if they could afford a horse, used them more for work than for pleasure riding.

I ask because in several JA novels characters are mentioned as being able to ride, but characters whom I thought should be riders are not mentioned on a horse (does Emma ever ride a horse, have I missed it?).

to my recollection, it is not mentioned; it IS mentioned with Elizabeth, Jane, and of course Fanny Price and Mary Crawford. I supposed it is greatly to do with preference - Mr Knightley, for example, prefers to walk.

Also, was it more common for men to ride and women to use carriages?

I think that especially in a large party of friends, the ladies would be in the carriage and the men would ride alongside. I think, also, it is more frequent for men to ride alone; women always had to ride accompanied, or perhaps preferred to.