The short answer is that when you throw a ball in a centrifuge in space, it travels in a straight line until it hits something. It experiences no pseudo gravity and effectively is not experiencing any acceleration.
The problem is simplified somewhat by evacuating the air so it's a vacuum. Lets say the surface of the centrifuge is moving at 50km/h, you're standing on the surface, and you can throw a ball at 50km/h.
First you drop the ball - simply releasing it from your hand. The ball will fly forward in a straight line at 50km/h, you are also moving at 50km/h, but will be pushed up by the surface of the centrifuge. This will create the visual illusion the ball is falling even though it's actually you being pushed up.
Next you throw it backwards at 50km/h so it has zero net velocity. What happens? The ball stays right where it is when you threw it, hanging in space with no force acting on it. If you stay standing where you are on the surface of the centrifuge, next revolution you will smack into the ball at 50km/h. From your perspective, it will look like the ball flies away from you at 50km/h, flies around the centrifuge, then smacks you in the head - even though it was actually just hanging in space. This is where a centrifuge deviates most absurdly from actual gravity, in a rotating cylinder, the illusion of gravity is caused by the object moving in a straight line until it intercepts the floor (which is curving up), an object with zero net velocity will simply hang there as the floor of the cylinder spins under it.
So if you throw the ball at 50km/h in the direction of motion, then it will travel in a straight line at 100km/h and smack into the surface of the centrifuge due to the curvature of the cylinder. The trajectory is actually no different to throwing the ball at 100km/h in a non-rotating cylinder, in both cases the ball will travel in a straight line, getting closer and closer to the floor, until it hits it. The apparent trajectory is not a parabola, as it would be with gravity, but instead a segment of a circle, which can however be a good approximation for a parabola for small angles.
When you add air, you get air resistance which will act to generally slow the ball, and air will be swept along with the surface of the centrifuge, effectively making a wind that will carry things with it. Air pressure will tend to push less buoyant objects to the surface of the centrifuge, and lighter than air objects to the center.
Exactly whether you can throw an object across the centrifuge to the other side (or "around the world"), depends on being able to overcome the velocity of the surface of the centrifuge. If the surface was moving at 100km/h, then a ball thrown at 50km/h will always fly ahead in a straight line and smack into the floor. The larger and faster the centrifuge, the more closely it will emulate gravity.