Sounds like fun! I'd say jumping would be the first english 'sport' I'd do if I rode english (I do western) and possibly eventing as second lol.
Do you want to dressage with her? I know one horse at my stables that was bred to either do jumping or saddleseat, I can't remember which now, but he turned out to be bad at it and ended up being my trainer's western horse.
I would like to do very basic dessage with her. I am not really into showing yet as my riding pony is only 4 1/2 years old and still learning the basics but I love to jump and hope to get more into showing this summer