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Haven't read the rest of this thread yet... because I am lazy, but...
Assuming no air resistance, you would oscillate about the center of the Earth. The gravitational force acts like a Hooke's Law force, since the force you feel is proportional to your position: (distance from center, i.e. r)
If there were air resistance, look up damped oscillators. In one damped oscillator situation, you would oscillate about the center with progressively smaller amplitudes, until you stopped altogether at the center (amplitude 0). In another damped oscillator situation, where the air resistance is really strong, you may not even make it past the center once, you'd just come to a stop. But that all depends on the strength of the air resistance, which I don't think in this situation would be that strong.
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