Rear hub assembly becoming loose, not allowing backward spinning
Question 1: I believe the question is how to tighten the lock nuts against the cones so that they do not loosen up.
The key to correct adjustment is a thin wrench - a cone wrench - that allows you to firmly hold the cone while tightening the lock nut against it. This question has an answer with a nice video from Park Tool showing how to work on a hub similar to yours.
Here's a picture of a mechanic holding the cone with one cone wrench while tightening the lock nut with another.

Question 2: What parts are missing that could be causing problems?
Swifty has identified that the pawls are missing from the picture and you have noted that you do have them along with the retaining wire. It sounds like your key problem is not having a cone wrench to get the cone/lock nut combination good and tight. It is difficult for me to know what might be missing without a diagram of what that hub should look like. Off brand hubs often don't publish diagrams. I couldn't find a diagram for the &quot;KLS Stormer&quot; hub (according to the linked catalog) that is on your bike.
Question 3: Is it possible to replace the cup and cone with sealed bearings?
It is possible but not easy. You would have to replace the hub with a different one (unspoke the hub, find a hub that works with your spokes or get new spokes and build the wheel) or use a different wheel.