Why not mount it flush with the exterior?
If you install an exterior outlet it needs to be weathertight. The box needs to be like the one shown, with caps on any openings, and it needs a weathertight cover. It also needs to be GFCI protected.
The grounding part is an interesting thing and I don’t have an answer for that. I can only suppose that in all instances, the bus itself is ground. Maybe someone can fill in that detail.