Does artificial gravity based on centrifugal force stop working if you jump off the ground?

Now, the question: if, while being inside a rotating space station, astronaut would jump really high, wouldn't he then experience zero gravity until he again will touch some part (wall or floor) of the station? Am I missing something in my understanding?

Well, here's a related question. Suppose you find yourself in an elevator at the top floor of a skyscraper when the cable suddenly snaps. As the elevator plummets down, you realize you'll die on impact when it hits the bottom. But then you think, what if I jump just before that happens? When you jump, you're moving up, not down, so there won't be any impact at all!

The mistake here is the same as the one you're made above. When you jump in the elevator, you indeed start moving upward relative to the elevator, but you're still moving at a tremendous speed downward relative to the ground, which is what matters.

Similarly, when you are at the rim of a large rotating space station, you have a large velocity relative to somebody standing still at the center. When you jump, it's true that you're going up relative to the piece of ground you jumped from, but you still have that huge tangential velocity. You don't lose it just by losing contact with the ground, so nothing about the story changes.

If you jump then you are in free fall, apart from air resistance, so you are weightless. This holds for any jump. For a brief moment, you experience zero gravity!

As I understand it, in order for the centrifugal force (which is responsible for creating gravity, in this case) to work, object it works upon should be attached to the wheels 'spoke' or 'rim'.

Not exactly. Centrifugal "force" is what's known as a "pseudo-force". It's the result of analyzing events using a non-inertial reference frame. If you're on a merry-go-round, and you treat the merry-go-round as being stationary while the world revolves around it, you will find that objects have a tendency to go towards the outside of the merry-go-round. So within the "The merry-go-round is stationary" point view, you have to posit some force pushing the objects away from the center, which is the centrifugal force.

But this force doesn't "really" exist: when you analyze the situation from the point of view of someone not on the merry-go-round, the objects are traveling in a straight line. It's just that any straight line will necessarily go away from the center. (Draw a circle, then draw straight line next to it. Imagine traveling along that line. From the point of view of the circle, you're first getting closer, then traveling away.)

When something is rotating, its velocity is constantly changing: although its speed is constant, the direction is changing, so the velocity is changing. Changing velocity means acceleration, and acceleration means force. This force is directed towards the center. Imagine driving around a circle counterclockwise. If you were to let go of the wheel, you would fly off the circle. You have to constantly turn left to stay on the circle. So there is a force, but it's towards the center of the circle, and is called the centripetal force.

From an inertial reference frame pint of view, a force is needed to stay on the circle; the centripetal force. But from a circular motion reference frame, the object is stationary. So if there's a centripetal force pulling the object in, there must be another force, the centrifugal force, pushing it out. So if you're standing in a rotating space station, you'll going to feel a force of the floor pushing you "up" towards the center of the space station, and since it feels like you're at rest (the space station is moving with you), it's going to seem that there must be some force pushing you "down" into the floor.

The important point here is that the contact with the floor provides the centripetal force, but the centrifugal force exists in your reference frame regardless of whether you have contact with the floor. Go back to the example of driving in a circle. Suppose you drop a ball in the car. Before you dropped it, it was moving with the car, and so just as the car had a centripetal force keeping in circular motion, the ball had a centripetal force on it. But for the fraction of a second that it's in the air, it doesn't have the centripetal force.

For an outside observer, the car is turning left, while the ball is moving in a straight line. The car accelerates to the left into the ball, and when the ball lands, it is to the right of where it was dropped. For someone in the car, however, it seems like the car is stationary, and ball is accelerating to the right.

Similarly, if you were to jump in the space station, then since you are perceiving things in the space station's reference frame, it will seem like you are accelerating towards the floor. This apparent acceleration exists regardless of whether you're touching the space station. That you are accelerating in the station's reference frame doesn't require physical contact with the station because it's not a physical phenomenon. It's simply an attribute of the coordinate system.

All of this applies locally: if you jump up, your motion will, in the station's reference frame, on small scales be the same as if you were being pulled down by gravity. This is an approximation that breaks down as you go to larger scales. These deviations from the approximation show up as other pseudo-forces, such as the Coriolis force. So being in contact with the floor does matter in that it keeps you moving with the station and reduces these deviations.