Where are Moon's Apogee and Perigee? Do they rotate too? 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?
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).

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