[dredmorbius]

On the "centre of Earth's gravity well" bit: what is the net effect of the fact that inside a sphere, the net gravitational attraction depends on the amount, distance, and vector of the gravitational mass in question, and that whilst inside a sphere at any point beneath its surface the net gravitational attraction varies.

In the case of the Earth, it actually increases to a certain depth, then falls as one approaches the centre (or would if one could; imagine, say, the neutrino version of yourself). My first thought was that the associated time dilation would follow a similar trend, though thinking this through, I'm not certain, though I'm leaning toward your answer above as not being accurate.