Yes, the force between the magnet and the cooper pipe is proportional to the relative speed, so you can leave the magnet still and move the pipe and get a force that cancels the gravity.

But in a real experiment, I think that it will be very difficult to keep the magnet in the right position, it will rotate and bounce against the walls, and the force will vary, so the average force has to be equal to the gravity. So your experiment is ideally possible, but I think that it will be very difficult to implement.

(In the original experiment, after a few moments, the magnet falls at an almost constant speed. This is the same speed that the pipe should have in alternative experiment.)

How 'bout with 3 rotating rings below it, orthogonally oriented, with a PID controller or something to change their speeds as appropriate?

Yes. You could expect it to bounce off the walls of the pipe, though.

The opposing fields will center the magnet.

Wouldn't that imply that the (presumably static, making a steady-state assumption) fields had non-zero divergence at the stability point, though?

No, what you get (assuming a constant diameter of the tube without dents or shape changes) is that the magnet will induce a stronger magnetic field in the wall that is closer which will push the magnet towards the middle.

It will overshoot a bit (but not as much as the original deviation) and so on until it goes smoothly down the center of the tube surrounding it without touching the walls.

Likely there will always be a slight wobble but eventually that should become too small to detect.