I wouldn't leave the door open unless the run was covered on the sides and top with sturdy 1/2" hardware cloth. I also like using wood screws to attach it, as nails or staples can be pulled out. We usually use washers with the screws.
Make sure there are no gaps anywhere and that the framing is sturdy. We use screws for that, too. Make sure the door can't flex and be pried open at the corners. I would also lock it.
It's good you thought about preventing digging. You can either bury wire or use a wire apron around the bottom, to prevent digging. Don't forget to make sure they can't dig in under the door. Sometimes people forget that area. I think the playhouse coop plans call for wire for the floor of the run area, but I wouldn't do that unless I covered it in something like sand and also gave them a nice, big, deep dusting box.
Our current run is ok for daytime use, but not secured for at night. We have an automatic door for the chickens, with a light sensor. For the chicken tractor we built as grow out housing, we went with hardware cloth.