Don't be cheap on the skirting, a digger is used to digging thru hard packed ground so keep that in mind. It doesn't have to be hardware cloth on the skirt, just about any welded or woven wire with holes 2x4 or less will stop them. If you do use hardware cloth, it never hurts to have some extra stashed away to repair or cover a additional vent or problem area.
Why hassle with digging in wire? Why not just attach one edge of the wire to the run and let it lay? It disappears as the grass grows thru it and can be mowed over it. It stops digging at the surface rather than after a hole is started. Wire laid on the ground allows you to lift it to move the run if you decide on a better spot for your birds. Laying the wire on the ground covers any low spots in the yard between the bottom rail and the ground that create inviting gaps. Why dig up the ground to bury wire and loosen it for the critter, making it easier for them? And finally, it's hard work burying wire and creates a muddy mess when it rains untill the grass grows back.