Can playing with the carotid arteries by compressing them at the carotid sinus(for the fun of the feeling) lead to carotid sinus sensitivity syndrome?

Not sure: I am not aware that gentle pressure on the carotid arteries causes any problems unless there is underlying disease like carotid hypersensitivity, or plaque. However recurrent forceful pressure can cause trauma to the arteries and cause a multitude of problems. It is better and safer to check the radial pulse in your wrist. You should not compress both carotids at the same time!