I'm wondering why you're posting this under butterfly, since I'm not aware of any reason for butterfliers to practice dolphin kicking on their backs.
Is your goal to improve your starts and turns when doing backstroke? If so, be aware that the dolphin kick used in backstroke is not exactly the same as the body dolphin which constitutes the core body movement of butterfly. Simply put, the butterfly body dolphin is initiated by pressing down with your chest and letting the water push your chest back up again, thereby creating a ripple that travels down your body to your toes. But the dolphin kick used for backstroke starts and turns is initiated by thrusting your hips up and down, and the result is that there is very little up and down movement in your body above your belly button.
If your goal is to improve your backstroke starts and turns, then I'm not sure why you're practicing it on the surface (since this isn't how you'd do it in a race) or why you're concerned that water is splashing in your face.
For underwater dolphin kicking, remember that your body will tend to follow your arms. The natural tendency of your body will always be to surface (because of the air in your lungs), so you will stay under by countering this. But if you overdo it, you can end up gliding toward the bottom of the pool.