There is no need for a foundation under the slab at the door detail. It could increase the risk of the slab cracking at this point. You are better off building a slab thickening toe at the point of impact, directly beneath the doors.
The trench only need be 600mm wide for a standard 300mm wide cavity wall.
Would this be the case for both doors. If I had a 4m door on the gable and 3m on the side?