If the moon had rotated on its own axis, would the moon have been a mini-Earth?

Answer

At first: the moon revolves around its own axis! The rotational period of the moon is equal to the orbital period, so that the moon always looks at us with approximately the same side. But she does turn! It is the tides of the earth that reduced the rotation of the moon, which was initially faster, and eventually synchronized it with the orbital motion (I already explained the correct mechanism in answer to an earlier question).

Second, the moon is NOT a piece of the earth. The moon came into being as a result of a collision between the earth and an external body, from which the moon was mainly formed. This aspect was also explained in response to an earlier question.

The cooling period of a planet mainly depends on its size. It takes longer to cool a large reservoir than a small one. Therefore, the moon cooled down earlier than the earth.

Finally: the formation of the atmosphere has nothing to do with rotation. The fact that the earth does have an atmosphere and the moon does not, has to do with the difference in mass. Air molecules tend to escape; the earth pulls on it harder and thus keeps it more connected to it.