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As the Earth wobbles during rotation, does the higher gravity at the equator tend to pull the moon toward an equatorial orbit even as the earth does that thousands of years wobble cycle? It would seem to me that the higher gravity, due to the larger diameter at the equator, would keep the Moon's orbit close to the equator. Or does the orbit of the moon with respect to the earth stay relatively stable with respect to the solar plane around the sun?

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The question is roughly equivalent to asking if the moon orbits in the plane of the ecliptic or in the plane of the Earth's rotation...I have a vague memory that the answer is in the ecliptic, but can provide no reference. –  dmckee Mar 4 '12 at 19:40

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The Earth and moon orbit around their common barycenter
I don't think the differential torque of the equator has very much effect on the plane of the moon's orbit - it takes a lot of energy to move the plane of an orbit!

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