How do you plan on a perimeter seal around a barn door setup? Obviously you'll want a screen door to keep out bugs and critters. That may be a difficult task given the hardware usually provides significant space for the door to roll past adjacent walls and trim surrounds. It also allows for the door to swing freely at the bottom. So wind may be a factor in keeping the door flush with the seal. For material, I'd prefer to use poplar vs pine and either build the thickness to 1 1/2" by laminating or source thicker lumber. Using 1by will be too thin and flimsy for a screen door. Thicker material will also make insetting the screen easier. It'd probably put the screens into individual frames that inset into the door frame so they can be removed to replace damaged screens, which is inevitable. Just some thoughts to consider.
Edit: Oh, the doggie door will also cause the door to swing out when the dog goes out...well if it's a medium to large dog.
Edit: Oh, the doggie door will also cause the door to swing out when the dog goes out...well if it's a medium to large dog.