Gravity needs mass to happen, right? So, theoretically, at the center of the Earth, mass is equivalent all around, and one would seem to be weight-less, right? In other words, the closer you get to the Earth's center, the less one should weigh. So, with that in mind, why did iron settle in the earth's interior during the initial Earth formation? Gravity would have drawn iron down initially, but due to the surrounding mass, that attraction of gravity would have been diminished, right? Where am I wrong in this discussion?

But it still would have been drawn down, and being denser, it would have been drawn down more than other lighter material.

Gravity needs mass to happen, right? So, theoretically, at the center of the Earth, mass is equivalent all around, and one would seem to be weight-less, right? In other words, the closer you get to the Earth's center, the less one should weigh. So, with that in mind, why did iron settle in the earth's interior during the initial Earth formation? Gravity would have drawn iron down initially, but due to the surrounding mass, that attraction of gravity would have been diminished, right? Where am I wrong in this discussion?

Two things. One is that gravitational acceleration reaches its maximum at the core-mantle boundary, 10.68 m/s².

 

More important, however, is that force is not quite the right metric. Energy provides a much better perspective. There is a huge difference in energy between a protoplanet with a more or less uniform composition and a differentiated planet with dense material mostly segregated in the core.