We often use 2x4 "no climb" fence wire for the runs / yards.
It's a better gauge of wire, so it's good against dogs. Daytime predators are different than night time predators. I don't worry about daytime racoons because the chickens can see them and get away from the fence. Hardware Cloth, IMO, is for the coop. The coop should have the quality skirting on all sides, while the run should work in tandem with the chickens good sense... size is a good way to work "with" because it allows them to get away and choose their area, or to filter back into the coop easier. I try to avoid making any features with feeders / waterers / other clutter inside the run which would be difficult for chickens to pass on all sides. To test, after chickens are installed, just go out and act like you're going to catch them, see where birds get hung up. Shift things around until they can easily flow around objects. Just my two cents.