How many chickens do you want to size the house for?
Chickens digging to get out (or more accurately, digging a dustbath hole that just happens to be so close to the fence that they can pop under it) isn't so much of a problem as predators digging to get IN. Anything that keeps out predators will most certainly keep the chickens in. Recommend either trenching in something impervious (heavy gauge wire mesh, aluminum flashing, big square pavers set on edge) or running an 'apron' out a couple feet from the fence (heavy gauge wire mesh set flat on ground with pavers or big rocks on it, or other similar arrangements).
You will kick yourself if you make a large run too short for you to walk around comfortably in; thus I would suggest 6' for height. A predator-safe top (a real roof, or heavy gauge 2x4 wire mesh supported by strong rafters) is the best thing but obviously also costs a bit more than weeblier solutions. Netting (deer net, trellis netting, bir netting, aviary netting) hung over the run, supported by a 'tent pole' or rafters, will keep chickens in and hawks out, but will NOT deter raccoons etc and will probably collapse in the first good snowfall or icestorm. Some people just run twine, with or without CDs hanging on it, crisscrossed all over atop the run for similar purposes.
Clipping wings varies in how effective it is in keeping chickens on the right side of a fence, and even if you get it so it's basically 100% effective, it does nothing for protection against predators which is probably the bigger issue. Even in cities there are LOTS of things that want to eat chickens -- hawks, raccoons, stray dogs, coyotes, all sorts of things.
Good luck, have fun,
Pat