Pop Quiz!

I heard someone's cellphone go off, so it's time for a pop quiz. Or if you prefer, think of it as a thought experiment. It's two questions. Answer me the best that you can, and explain your reasoning.

1) A tunnel is drilled straight through the Earth, straight from one side to the other and through the core. Let's suppose the tunnel is given strong enough walls that it doesn't collapse, and that it's well insulated so it isn't too boiling inside. A tourist sees this gigantic hole and decides "hey, there's no bottom to hit, so it's gotta be safe to jump into it," and follows word with deed. (a) Where does he end up? (b) If he could stop at the middle, what would be his weight there (as compared to his weight standing on the surface)?

2) Same tunnel through the Earth, this time you can walk along it with a barometer, reading air pressure throughout the tunnel. When you get to the center, will the air pressure be greater than that at the surface, the same, less than the surface, or entirely zero?