Imagine you're in a spacecraft that uses centripetal force to simulate gravity. What happens if you jump?

You’re in a cylindrical spacecraft that rotates around its axis at such a speed as to create 1g of acceleration, simulating gravity.

When you jump, do you land back on the same patch of ground that you left? Or does your angular motion not match that of the spacecraft if you leave the ground?

What about the air inside the spacecraft? Is it constantly windy because it is not rotating with the floor?

If you get on a ladder and climb closer to the center of the spacecraft, the “gravity” gets stronger, correct? But if you reach the very center, there is no force.