I have a rear wheel that has needed work 3 or more times in around 3 years (of not very regular riding) and I'm wondering if something is especially wrong with the setup on this FUJI road bike.

A few years back I reworked the axle on the bike - put some grease in it and set it. Later it had come very stiff. I opened it up and bought some cones to replace it - fairly chewed up.

This year I took it to a cycle-hobby shop as the wheel was wiggling like out-of-true, but no one portion of the wheel consistently rubbed. Sure enough, they say it needed bearings, now it trues up perfectly and all good?

Now, after a few rides and a few months, I'm seeing it is obviously loose and furthermore the little round washer was coming loose. Pushed it in and rode a bit, and this part is still not completely fitted in as I can push it in a little.

What is that part called, and is it safe to run with that a little loose? Is just tightening the cones and tightening/loosening likely to fix this for good or... for another year at least?

The chain goes forward just slightly when freewheeling but apparently that is supposed to be normal? Not sure if it is in any way related.