The axle is very unlikely to be bent, and even if it were bent, it would not cause the wheel to run out of true.
The wheel rim can usually be correctly centred by tensioning or loosening certain spokes as required to get the rim running perfectly concentric with the axle and laterally true.
The wiggling freewheel is not bad enough for me to be concerned about it, and it may not be possible to improve/cure it without changing side cover/s and possibly the motor ring too.
It would be nice to know what was causing the jamming sound, and what exactly it was that you tweaked to improve it?
I'm also a bit confused as to what could have "jammed twice" at higher throttle to cause the huge braking force.
If the rubbing noise disappears when you pedal
(like the noisy freewheel bearings in the update 2 video) it will be a problem within the freewheel itself. Replacing the freewheel may cure the noise, but it is unlikely to make any noticeable difference to the "wriggling".
If the rubbing noise is still audible while pedalling, it could be the side cover rubbing against the black protective sheath on the Hall sensor wires inside the motor:
You can just see the circular rings on the inside of the cover of my MPIII where the cable had been gently rubbing against it:
I simply drilled a small hole in the stator to allow a cable tie to be used to hold the cable tight against the stator, which prevents it from rubbing against the inside of the side cover:
This is what it looks like from the other side of the stator
(with the access plate removed).
Your motor seems to be rotating quite freely in your video with no obvious signs of binding.
Alan