No, it doesn’t. It moves a little bit, but it does not rotate.
Yes, the moon does rotate. It's rotational period matches its orbital period - that is referred to as tidal lock. If it did not rotate, we would see the entire surface as it orbited the earth.
Try an experiment with a couple of balls - mark each one and then move one in a circle around the other, keeping the marks lined up. You'll notice that the ball in "orbit" completes one rotation about its axis for each "orbit" it makes around the other ball. The try it again, keeping the mark on the "orbiting" ball lined up with a point somewhere else in the room. You'll notice that the orbiting ball continually presents a different side of itself to the ball it's circling as it orbits.
The same thing is happening to the Earth-Sun system. The Earth's rotation about its axis is very gradually slowing down, and eventually the Earth will be tidally locked to the Sun and one side of it will always face the Sun while the other side will face away. But that won't happen for a few billion years.