Is my rear cog wobble the fault of the axle, the hub, the bearings, or the wheel trueing? Adjusting the rear derailleur on a Bicycle Shaped Object in a stand, I observed skipping gears when shifting from 6 to 5 (it skips to 4), 5 to 4 (it skips to 3), etc, but would snap in place correctly when going in the other direction.

Observing this, I then noticed that the cogset is wobbling. Maybe the hub is wobbling. And the rim is appearing to not be centered (visually, radius does not equal half-diameter) around the hub. Distance between brake pads seems fairly consistent for a BSO.

When spun softly, the wheel does not balance with the reflector down.

Is my rear wheel wobble the fault of the axle, the cogs, the bearings, or the wheel trueing? I suspect the skipping is ultimately a derailleur alignment issue, but I can't begin to work on that until I solve this.



Video of same:
I think you have a bent rear axle.

Looking at your rear block it does not have a lockring on the cassette, so its likely a freewheel  (ie the clicking part is integrated with the gears, not part of the wheel)

This means the axle is less-unsupported on the right hand side, and they bend easily.  

Here's a pic - the red circles are the ball bearings, and the freewheel axle (schraubkranznabe)  has a long lever arm of 2-5 cm compared to under 1 CM 



The upshot is minor bumps and regular riding are enough to deform the rear axle over time, right where the bearings are.

As such, the rear wheel oscillates because the bearing cone on that side is no longer equal distance from the bearing cup all the way around.

Your fix is a new rear axle for $20ish but it could happen again.   The longer term fix is to replace the hub with a cassette-based wheel. 

Or simply ride it like it is and keep your eyes open for another rear wheel.  The bearings will munch themselves over time, and the cone will get pitted and damaged, but as you rightly point out, its a BSO.