Well, the pressure from all the weight of everything on the Earth crushing you down would most likely kill you. In terms of gravity, you'd reach the Earth's centre of gravity, so I'm not sure. At a guess, if you stopped in exactly the right place, you'd hover, but it would have to be very exact. And since if you remove the physical core, the core is no longer the centre of gravity, its somewhat redundant.
The inner core isn't lava due to the pressure being exerted on it, the outer core is liquid but you wouldn't be there.
Perhaps you are getting at the hypothetical situation where you are in a hallow room-sized sphere at the exact center of the earth that is indestructible and well air conditioned. I'd imagine that everywhere would be down. That is, you could walk all around the sphere, and if you jumped then you would "fall" to the other end of the sphere. Just a guess.
I think that would depend on how large the sphere was.
I Think that you would have a large amount of iron and all the natural disasters at the same time, since the earth would collapse inwards with all what it come with, perhaps.
The earth would collapse inwards. Also, due to the lack of the iron of the core, the earth's electromagnetic field would cease to exist, leaving the planet vulnerable to solar flares.
If there was a hollow sphere in the center of the earth I imagine you would feel weightless. Just like those frogs suspended in the center of super magnetic fields.
in short, as you go below the earths surface, the effects of gravity drop off. The force that is exerted on you is the same as if you were on a planet with the radius equal to your distance to the center (assuming the planets average density is the same as that of earth). Note this is only the case for going underneath the earths surface. A point mass at the center of the earth feels no net force.
If you were somehow able to create a chamber at the exact center of the Earth capable of supporting the mass of the rest of the planet above it and preventing you from burning up from the intense heat, the most automatic assumption is that anything inside the chamber would be in free fall. However, this isn't actually what would happen; instead, there would be gravity in exactly one direction, and you would be attracted in that direction. The free-fall assumption is based on the assumption that Earth's center of mass and the center of the Earth are the same place, which is in turn based on the assumption that the Earth has a fairly uniform density and gravitational pull. However, as this graphical depiction of data from the European Space Agency's GOCE mission demonstrates, http://www.youtube.com/watch?v=d4E5T9PSbo0 Earth's gravitational pull isn't uniform, and therefore its density isn't either. This means that we can't just assume the Earth's center of mass is the exact center of the Earth, and we can't just assume the contents of a chamber at the center of the Earth wouldn't experience gravity. In order for an object in such a chamber to be in free fall, the chamber must be centered around Earth's center of mass.