Ball bearings only on one side, and smashed "ring"? I have been given a bike for free, as the rear wheel was very wobbly.

I'm been learning how to take things apart, and investigated the axle - it's not bent.

The ball bearings, however, were a surprise. There were 9 on one side (the side of gears) and none on the other side. Those 9 balls were kind of loose (I think 10 could fit well), and there was a smashed up steel ring in there.

I'm not sure if that's something "useful" (eg, reduces friction?), or if it's the "dust cover" that got pushed in and smashed thinner. I found it weird that only that side lacked a steel dust cover.

Could you help me understand if this was just a bad maintenance job (the rear wheel is clearly newer than the front one) or if it makes sense to have ball bearings only on one side?
There are supposed to be ball bearings on both sides of the hub. You need to rebuild the bearings replacing at least the non-drive side balls. 

Bicycle hubs with loose ball bearings use a ‘cup and cone’ system. The cups and cones have the bearing surfaces and the balls are sandwiched between them. The cups are in the hub, the cones are threaded into the axle. 

There are many videos on YouTube that will show you how to disassemble, repair and reassemble a cup and cone hub. RJ The Bike Guy has a good one. 

You’ll need to inspect the cup and cone bearing surfaces for wear or pitting. the axle, cones, spacers and balls can all be replaced. Balls come in a couple of different sizes. If you don’t have accurate calipers a local bike shop can help you out.