Can I take the car to the northern, occupied Cyprus?

No, because of few reasons: INSURANCE (no insurance covers you across the Green Line), SAFETY (half of cars over there are with the steering wheel on the left, NO BREAKDOWN service available). If you will go there anyway please note that thee is no insurance cover, nor road assistance. If anything happen there you will be entirly responsible for any damage, towing etc and other costs that may arise.