If gravitation decreases as we approach the core and if density is a function of gravity we should expect a void core.
Unless the accretion disk was formed starting with the objects of highest mass available first.
But as soon as the whole planet would have been liquefied the core should have voided according to 1.
Maybe there is some confusion between weight and density / specific gravity. The weight = mass x acceleration. True, the gravitational attraction is zero at the centre of the earth, but everywhere else there is some gravity acting on a huge density contrast. Also there is the question of incompatibility between siderophile and lithophile elements. During the initial hot phase of planetary formation it wouldn't have taken much gravity to differentiate the two liquid phases. That said, it is possible that some residual non Fe-Ni lighter elements could be lurking within the core - sulphur and hydrogen (as metal hydrides) have been widely cited as possibilities. The main evidence being that the true density, as determined by geophysics, is slightly less than it should be if it it was just nickel-iron (+ a few heavier elements).
answers the question as posed in the title. Matter deep in the Earth has the weight of all the matter above pressing it down. This post explains how high that pressure can get.
Although it is true that gravity is zero at a single point at the very center of the Earth, where gravitational vectors completely cancel each other out, it is never negative. Nothing is pulling matter away from the center. There can be no void. Premise 1 in the body of the question is simply wrong.
Furthermore, everywhere away from that single point, there is a little bit of gravity left over from the vector sum. The direction of that result vector is always towards the center (given some allowance for the Earth's equatorial bulge).
The micro- (and even zero-) gravity at the center of the earth is more than made up for by the pressure of the entire Earth holding it in place, trying to squeeze it down to a point.
This was true even when the Earth was molten, so the heavier elements tended to sink down to the center. This other post has a thorough explanation as to why.
Now that the Earth has cooled somewhat, pressure deep in the Earth increases local density by forcing liquids (e.g. the liquid outer core) into a solid state (the solid inner core), and then as the pressure increases, solids shift into new, more tightly packed crystal structures.
Gravity is a weird force.
All mass has, and is affected by, gravity. That means you, me, your cat, black holes, the sun and the earth.
It also means each little bit of the earth is attracted to every other little bit. So every bit tries to get as close to every other bit as it can.
The shape that these bits form is a sphere - because every bit is as close as it can get to all the other bits.
Now you are right - an earth which is matter all the way through would attract in the same way as one which had all the mass concentrated around the edge. But we know that the centre isn't hollow for two reasons:
Volcanoes and Earthquakes. Volcanoes erupt liquid rock from the ground, which shows that there is something down there. And we know that there is something because we feel the shaking from earthquakes on the other side on the earth - and we can map the layers of solid and liquid from these.
- If gravitation decreases as we approach the core and if density is a function of gravity we should expect a void core.
No, because every bit wants to be close to every other bit. It doesn't just want to be near the bits on it's side, it wants to be near the ones on the other side, and so the "void" would close.
Why is the density of Earth higher at the core?
The denser stuff sinks. This is how it works everywhere that gravity applies. We can explain why.
The denser objects have more gravity than the less dense objects. This means all the objects are attracted to the more dense objects more than they are the less dense objects. Here, red is more dense.
But the ones over the other side, are also attracted to that dense object, so this happens:
Eventually, you end up with all the less dense objects surrounding the more dense object, as they try to be as close as they can to it. But the more dense objects are attracted more to the other dense objects (as they both have more mass). This means all the dense objects clump together, surrounded by the less dense ones.