I have some questions about the expansion of the universe:

1. The expansion rate is almost 70 km/s for 1 Mpc, so lets say we have a rigid cylinder having a length of one Mpc. As far as I understand, the two extremities of that cylinder will have an "absolute speed" of 35 km/s in direction of the center of the cylinder. Is that correct?

2. If that's correct, that would imply that a cylinder can't have a length greater than 8571 Mpc because that would imply extremities speed greater than light speed. Is that correct?

3. If that's correct, that would imply that two objects or particules distant of more than 8571 Mpc can't get closer. Is that correct?

4. If that's correct, that would imply that we will never be able to "see" an object distant from more than 8571 Mpc. Is that correct?

What you have calculated is the diameter of the Hubble sphere. The (average) recession velocity relative to us at a distance $d$ is given by:
$$v = Hd$$
where $H$ is the Hubble constant, and you've calculated the distance at which the recession velocity is equal to the speed of light. Beyond this distance objects are moving at a velocity greater than the speed of light away from us.