The Moon cycles every 29.53 days. Not because of Earth’s shadow, but because our viewing angle on its sunlit half changes.
A common misconception. Phases have nothing to do with Earth’s shadow. Half the Moon is always lit; what changes is the angle from which we see that lit half.
Longer than the Moon’s orbital period of 27.32 days, because Earth also moves around the Sun in the meantime.
The Moon rotates once per orbit, so the same side always faces us. This is tidal locking.
Each phase rises at a different hour. A crescent appears low in the west after sunset; a full Moon rises in the east as the Sun sets and stays up all night.