Axle - rear hub and washer thing a bit loose
The round washer-like piece that's loose is a metal dust cover. It keeps debris and water out of the bearings. On some hubs they're pressed into the hub shell and on others they're pressed onto the cone, which is what this looks like. Having it be loose is consistent with replacing the cone with one that wasn't quite the right diameter to mate properly with the cover. There's not much leeway there.

The usual procedure for transferring the dust shield is to cradle it with a vise or socket and very gently tap the old cone through and the new one on.

You don't really want it to be loose because they're the hub's main line of defense against contaminants. If it's rotating around freely it may get damaged as well, and be unable to press into place on a new cone.

Replacing cones is a careful business of finding one where the ground part matches the profile of the original one well, making sure the ball track is falling in a good spot on it in practice, in this case ensuring it has the right major diameter for the dust cover, and then making sure the spacing of the hub is right when you're done. There are some in-depth treatments out there of this, Barnett's Manual has a good one, but it's a fairly big topic. Repeat problems with the hub coming loose, rapidly accumulating wear, etc are likely outcomes of the match not being great with the profile of the new cones. The balls may be making contact in an awkward spot of the curve or off of it completely. An extreme case of this may make the wheel not usable. It could be a good idea to look at the cones and see how they're wearing.

It's also possible that the cones were a good match but the cups are damaged and causing repeat loosening. If so that's likely to keep happening but take a long time to be an acute problem.

The chain behavior is probably some dryness or gumminess in the freewheel and not related.