About falling though a hole in the earth, you are correct -
"Well, as the ball drops through the hole it picks up speed - that's the acceleration due to gravity. It reaches its top speed at the very center of the globe and continues rocketing toward the other side. Now here's the tricky part - because gravity is tugging on it, as it heads for the other side of the world, it starts to slow down, at first a little bit, and then more as it approaches the surface.
And just as the ball reaches the surface it comes to a complete stop, but just for a moment. Then it falls back down the hole and the process repeats over and over and over, like a kind of intra-planetary pendulum but with no strings attached."
- David Stevenson, professor of Planetary Science at Caltech