I The gravitational center of the Earth

[qnt200]

We know that gravitational forces are nullified near the center of the Earth, so the gravitational field's influence is not felt. Is it because of the Moon's gravitational field that the area of zero gravity has shifted away from the center of the Earth? If this is the case, this eccentric area will orbit around the center of the Earth due to the moon's orbit.

 

[Orodruin]

The gravitational field is never ”felt” in free fall regardless of where you are. Whether the gravitational forces equate to zero depends on your frame of reference.

If you want to consider the Earth-Moon system, them the center of mass is indeed not at the center of the Earth.

 

[Doc Al]

The Earth and Moon orbit about the center of mass of the Earth-Moon system. This point, called the barycenter, is about 1,700 km below the Earth's surface.

 

[Bandersnatch]

However, it's important to stress that an object placed at the barycentre would still be attracted towards the centre of the Earth. I.e. the fact that the Earth and the Moon orbit their mutual centre of mass doesn't affect where the forces produced by the Earth on itself cancel out. This follows from the point Orodruin made - everything on/inside the Earth is in free fall w/r to the Moon, but not w/r to the Earth itself.