If magnet was not rotated when dropped through the tube, would it simply stick to one wall in case of slightest imbalance?
No. Because copper isn't ferromagnetic, it will not be magnetized by the magnet, and consequently the magnet and the copper will not be attracted to each other. (In fact copper is diamagnetic, so it will actually slightly repel.) Source: I am a magnet scientist.
Judging entirely by the name, that sounds awesome :D What do you do all day?