I think elbow patches are just reinforcements on jacket of any style that has worn out on the elbows, where a jacket will get particular stress. Though you sometimes get them already in place on new jackets, this is more of a fashion thing. Traditionally they are just a way of making a good jacket last longer. Though it's not something you would do to a formal jacket, so yes it's more associated with various kinds of sports or casual jackets in hard wearing fabrics such as tweed or corduroy.
A hacking jacket is for hacking (riding a horse) and a shooting jacket is for shooting. Sounds obvious, but the point being that each will have certain characteristics relevant to that sport, eg a shooting jacket may have pleats allowing for ease in raising the arm, hacking jackets have splits/vents designed for sitting in the saddle. Both are a type of sports coat, in the sense of country sports.
This is my understanding of the terms anyway. You often see sellers using terms like hacking and shooting on jackets that aren't true examples of the style.