The end velocity in the example I wrote about was if you did take spin into account but still had the absence of air friction. If taking spin into play, your angular velocity exiting the hole would be very fast. The example you gave for the idealized model would be completely correct. You would exit the same speed as you went in and go only to the height you jumped in at.

I think for air pressure at the middle of the hole, or the center of earth, the scientist say it is about 3.6 atmospheres. The way I try to understand it, think of it as the column of air above you is pressing down, and the column of air from the exit side of the hole also pressing down at its max while you’re in the middle. You’re going to have the maximum air pressure at the center (if any of this was even possible). So as you go to the other side the air pressure would then decrease exponentially? I add a question mark because this is most simple way for me to try to understand it. Great question.