I'd suggest LenX, rather than the native camera app - it allows time exposures and, more important, what used to be called Bulb exposure - just open the shutter, flash the bulb, then close the shutter. Set the phone up with the camera facing Polaris, open the shutter and wait a while before closing it. That's how you get those circular star trails.
Anyone can get a good picture of the full moon - the brightness is about 1 lux. A starry sky without the moon is about .0001-.002 lux for the entire sky. You want a very long time exposure. Remember, an arc of only 15 degrees of the stars takes a one hour exposure. Even an f1.5 lens isn't exactly a light-gatherer. Time exposure is.