Can you use a cooking stove at primitive sites? How do you cook if your staying in primitive site?

Yep, portable stoves are allowed but not provided so you'll have to bring g your own. Just remember to bring fuel (propane canisters or whatever) with you. You can also make a fire in the fire pits and cook over the fire if you're comfortable with that. Aluminum foil packets and hotdogs on skewers are easy to cook in the fire. The store there sells firewood as well as everything else you'd need (firestarters, matches, lighter fluid,etc) to get your fire going. Oh and I want to say they carry propane canisters if you forget yours (for the stove).

Yes

You can get a little propane stove or cook on a camp fire. There is a grill top on the fire ring