Bearings jammed up even though there is play
You've given the cause in your first sentence.  I've got a cheap new rear wheel.

Cheap wheels tend to be not as good as better wheels.  So the bearing surfaces are not quite as exact in their tolerances as a more refined hub.
Is it possible that you have swapped the cones side-for-side ?

At these scales, its impossible to see visually whether its the cup or the cone which is out.

Lastly - did you change the bearings?  Are they the right size?  Are they all the same size?   You might be better to simply replace all the bearings with new ones, that way they're the same batch and thus size.

Rear wheel hubs should be 1/4" bearings (about 9), and front wheels are 3/16" (about 10)   I'm guessing you have loose ones, rather than those in a retainer.  Either is fine, loose ones are more fiddly but technically better. 

The bearings looking like they are on top of each other is probably happening as you unscrew the cone, not while its spinning.  There is simply not room for a whole bearing to jump another one, even when its got a lot of play.