The 1/2" HC will prevent smaller predators from entering the coop/run. Just be sure that there are no other cracks or holes that are larger than 1/2"...around doors, roof eaves, windows, run corners, etc.,. As I said, dogs and large coons will go through the hardware cloth as it is no where near as strong as 2x4 welded or woven wire. To avoid the possibility of a predator breaking a weld in welded-wire fencing go with woven wire...no welds. Woven wire is more expensive, though, but it's good stuff. A large predator will have to break several welds to be able to get through 2x4 welded wire fencing.
If you opt to go with a run of 2x4 around the lower part of the 1/2" HC then I would purchase 5' wide 2x4 fencing. This way you will have enough to extend up the sides of the fence 3' while leaving 2' that can be bent outward for an apron around the coop/run. This won't stop rodents or other small digging creatures from entering but will stop the canines and coons.
If you've got the extra cash to cover the entire run and coop in 1/2" HC by all means go for it....the tighter the security the better you sleep. But, I would reinforce the bottom with something heavier. Ditto on not using staples.
Best wishes,
Ed