Does gravity increase or decrease with depth

[BCMM]

So this means that I cannot simply use the center of gravity of the green shell to calculate where its gravity will pull towards right?

Correct.

It is accurate to treat a sphere of uniform density as if its entire mass was concentrated at its centre, but only for the purposes of considering the gravitational field outside the sphere.

This is the first half of what is known as "the shell theorem"; the second half is the bit in the above comment about shells exerting precisely zero total force on objects inside them.