Where are Moon's Apogee and Perigee? Which one is inside, towards the Sun, between Sun and earth?
Do they rotate too just like the north, south node?
Does Moon always need to be on Apogee/Perigee for it to be Full/New Moon?
The full moon and the new moon occur when the moon is in line with the sun. Since the Earth is orbiting the sun, the position of new moon in the moon's orbit moves. The moon takes about 27.3 days to orbit the Earth (relative to distant stars) but a new moon occurs about every 29.5 days.
So the direction of the perigee and apogee (relative to the sun) change by a couple of days each month due to the orbit of the Earth, and a new moon can occur at perigee, apogee or any time between.
Also perturbations of the sun cause the direction of perigee (relative to distant stars) to precess. It takes about 8.85 years for the direction of perigee to change by 360 degrees. They rotate, but not at the same rate as the nodes.
This image, taken from wikipedia shows this. The wavy line shows the distance of the moon. The saw tooth line shows the phase of the moon; vertical lines indicate new moons. You can see that sometimes a new moon occurs when the moon is close (February) and sometime when it far (July), and sometimes in-between.
The Moon's distance from Earth and Moon phases in 2014.
Where are Moon's Apogee and Perigee? Do they rotate too?
Yes they do rotate!
Apsidal precession is the rotation of the line of apses (line connecting apoapsis and periapsis and passing through the Earth). Lunar precession takes this line about nine years to rotate once around the Earth, referenced to the celestial sphere (the stars).