Again, the present model for the formation of the Moon is that it's material was thrown off by the Earth in a collision between another body and the Earth.
The Asimov article you are referencing appears to be "Just Mooning Around", which was collected with other articles in "Of Time Space and Other Things."
All the articles in this collection were written between 1963-65, several years before we able to examine actual Lunar material. A great deal of information on the Moon has been collected since then and led to a good amount of additional thought on the origin of the Moon which has led to the collision hypothesis.
As to the Moon's rotation on its axis. While the Moon does complete one rotation in the same time as it completes one orbit, it does not rotate at the same speed as it orbits at all time. The rate of rotation is steady, but do the eccentric nature of it orbit, the orbital speed changes throughout the orbit. This results with the Moon sometimes rotating at a rate slightly slower than it orbits and sometimes rotating faster. So it rotation alternates between falling behind and speeding ahead. This libration isn't enough to let us see all of the Moon's surface from the Earth, but we do see more than 1/2 of it.
In addition, the Moon's axis of rotation is tilted slightly so that it does not align with the axis of its orbit. So, yes, the Moon does rotate on its axis.
As far as the dating of Earth and Moon rocks go. Earth is geologically active. Radio-dating of rocks can only go back as far as when that rock cooled from being lava. Since the rock cycle is continuously recirculating crust material, the oldest dated rock found on Earth only sets a lower limit for its age, not an upper one. The Moon lost any interior heat it had much sooner than the Earth, and its oldest rocks haven't had to deal with being recycled and having their radio-dating clocks "reset".