To attach hardwire cloth to hardwire cloth, I use J-Clips. You can get them and the tool you need to attach them at the rabbit section of
Tractor Supply. They are traditionally what is used to make all wire rabbit cages. Wire threaded through will also work great. I suspect hog rings will work OK too, you just need to squeeze them down real tight so you don't leave gaps something might squeeze through.
To attach hardware cloth to a flat wood surface, I use lathes. This is a strip of wood maybe 1/2" think. I predrill holes for the screws so the wood does not split and attach it with wood screws suitable for outdoor use. I make sure the screw goes through an opening in the hardware cloth and screw it down pretty tightly. That hardware cloth will not go anywhere. To me, one advantage of this over using the screws and washers is that the wood covers up the sharp edges of the wire. I don't mind the sight of blood, even my own as long as it is in small amounts, but I don't especially enjoy it.
To attach hardware cloth to posts and such that are not flat surfaces, I use fencing staples.
Raccoons, possums, and many other things can climb over a fence and do damage to chicken wire. You'd be surprised at how well foxes, coyotes, and many dogs can do getting over a fence. Chicken wire over the top will help add to the safety of your run, especially during the day, but it does not solve the predator problem. They can still get in, even during the day. The reason night is usually the more dangerous time is that many predators are more active at night and they normally have more uninterrupted hours to work on that wire.
Good luck. Hope this helps.