Play between axle and bearing inner ring
I know this type of hubs a Open Bearing hubs. You can have a "basket", a sort of ring that holds the ball bearings in position, and just ball bearings sticked with grease.

The way the axle sits on them, fully asambled should be like, from left to right: LockNut(quick release if you have one),dropout,Counter-Nut, Cone,hub, Cone, Counternut, dropout, locknut.

So, your problem is that you have play, but when you mount the wheel it goes stiff. 

For this problem you have to tight the CounterNut and Cone, together on one side. Then tight the other cone, just enough so you don't have play and runs smooth. Grab a thin wrench, if you have a cone wrench will be better, hold the cone with it and then with a second wrench tight the counternut.

Don't overtight the nuts, as a cheap axle is really easy to stripe. If the nuts keep loosing themselves get some preassure washers(the spiky ones) and put one beteween the cone and the washer.

cheers.

EDIT: Probably the black thing is just a dust seal, which makes seem like a sealed bearing.