I think you're definitely on the right track, switching from chicken wire and staples, to hardware cloth and wood screws with washers.
The other thing I would suggest, is that you add something to prevent digging. Once the dog can't get in by tearing through or pulling off the wire, it may start digging. You can either bury wire a foot down into the ground, angled slightly outward or attach a flange of wire on top of the ground, a foot out. Eventually, grass can just grow through the wire that lays on top and you won't even notice it.
The good thing about the work you are doing is that it will also protect your chickens from any other dogs that may eventual come by, owned or stray. It will also protect them from wild predators. Hopefully, now that you are going to have such a nice, sturdy run, you won't have any more losses or have to do any more run fixing.
I hope everything goes very well with you and your new chickens!