SFG boxes only need to be 6" deep, but I needed to sink mine into the ground due to the slope in our yard so mine are 12" deep. After almost a year of using them I'm glad I went deeper.
We tried keeping the grass between the boxes but it was so difficult to mow, so that went bye-bye and in came some weed control fabric and mulch. Weed control fabric does not, in my opinion, actually do what it claims. :)