Drywall screws are fine.
I stopped using drywall screws after having some break, and they are the worst for rusting and bleeding through. Unlike construction screws, they are not specified for live loads. I suspect the problem is that they are not heat treated to prevent work hardening, but maybe it is just a problem with cheap material selection.
IIRC, they're somewhere around 1045 steel, which doesn't really harden, but still hard enough that it tends to snap rather than bend. They're good for some strength, but to be fair I should've said
deck screws -- or I suppose that includes the "construction" kind -- essentially same style, but made for it, properly tempered (hm, I don't know what alloys, actually), and coatings to reduce friction and corrosion.
Main thing is, just avoid those traditional/old style tapered screws: mild steel, huge friction, half of them strip out before even getting flush, even with pilot holes. Man I hate those things...
It is not as pretty, but a strip can be added along the edge underneath a thin shelf to strengthen it preventing sagging under load.
Yep, greatly saves on material. The sides are braced by the shelves so don't need much any stiffening, but the shelves themselves benefit from a flange or beam along them. Doesn't have to be at the front (though, that may be the most convenient mounting place), and doesn't have to be wood either -- indeed, if you need to save vertical height, or weight, consider an exceptionally stiff material (in comparison), like steel! Just mount it in such a way to avoid expansion issues -- drill slotted holes for example. Hot-rolled flat (mounted flush to the face) or angle (mounted face or underneath) would do well, and has a soft corner safe for touching; it might be worth sanding and varnishing/painting before assembly to get a more pleasant finish.
Even aluminum is a great improvement, though you need more of it (although, less by overall weight, if you can afford the increased vertical space).
Tim