Rent a few first...rent a sit-inside and try it out. Then rent a sit-on-top and see which one you prefer. Better yet, get a buddy to go with and rent both kayaks at the same time so you can try them out in the same conditions, back-to-back.
I've seen people carry their dog in a sit-inside...but the ones I've seen were either tandem kayaks so they had an extra long cockpit opening or really small dogs.
As far as stability, they're similar depending on which ones you try. Hull shape and width make a difference on stability. Have you looked at the Native Ultimates or the WS Commanders? Kind of a hybrid between a sit-on and a sit-in and a canoe.