Random Shower Thought - Hollow Earth

Provided there was a perfectly straight tunnel right through the center of the earth... If one was to take a rock and drop it down that tunnel... would it fall all the way to the center of the earth, over shoot, turn around, fall back towards the center, overshoot etc. etc. until eventually coming to rest, floating weightlessly right at the center of the planet?

If so, what would be the force holding it there? Would it be gravity PULLING it with equal force from all sides as the mass of the earth would now be equal on all sides, pulling the rock towards it? 

If that were the case, the same force should be acting on the material of the earth itself as well. Does this mean there is a point somewhere half way towards the center of a planet where gravity and pressure are greatest and would they then actually start becoming less again towards the center of it? Add to this the centrifugal force of a planets rotation, actually pushing material outwards... Perhaps there really is a cavity at the center of the earth? 

There is a video on the internet of a rotating ball of water with air bubbles in microgravity and the bubbles do collect in the center of it. Admittedly this little sphere would not have much gravitational force to speak of so it's held together by surface tension instead. There are probably different physics at play at this scale...

Here is another video this time with a non rotating sphere of water.

Be that as it may ... the deepest we have ever drilled into the earths crust is the Kola Super Deep Borehole which is just over 12 km deep while the distance from the surface of the earth to it's center is 6371 km on average... In other words we have barely even scratched the surface and anything that goes on at the center of our planet is nothing more than a theory based on our current understanding of the universe which changes all the time.