You do not need to provide 0.5% reinforcement for such a small pad. That reinforcement ratio is for an infinitely large slab with no control joints.
You should design the pad for any loads imposed upon it. I typically create a 3D finite element model supported on springs.
If you can justify the load path, you are OK from a strength and stability point of view. However, the corners could open up and allow ground water to seep into the vault. This in turn could lead to sink holes forming at the ground surface.
Normally the walls in a below grade vault are two way slabs -- they span horizontally and vertically. Without corner bars, I think your walls have to span vertically, from base slab to top slab. Are there no dowels between the wall and the base slab?
When it comes to connections, I do not follow ASCE 7 literally. If you do, you need to calcuate the effective area to an individual fastener which makes the wind load unreasonably high.
If there is room at the edges of the base plate, you could use steel angles (5" vertical leg and 3" horizontal leg), with the vertical leg welded to the base plate and the horizontal leg connected to the pier with adhesive anchors.
If you use steel H-piles for your soldier piles, you could weld plates on each side of the web, creating a slot for the timber lagging to slide down inside. For the first excavation, the lagging would bear on the flanges. For the second excavation, the lagging would bear on the plates.
You should include wind load in the proper load combination from your code. Wind can cause suction on a wall, thus adding to the driving force from backfill. However, with load combination factors, the case with wind load may not govern.
I assume that means lateral-torsional buckling does not apply to that particular member size.
I recommend you check the design by hand to confirm this is the case.
I would have them design a portal frame by hand, a braced frame by hand, and a shear wall by hand. I would also have them do designs by hand in all four major structural materials - concrete, masonry, steel and wood. Have them design a spread footing by hand. And a cantilever retaining wall...
