You want the strong, silent type, which is characteristic-counterintuitively-of the lowest and thus best gauges. No need to lose sleep over 16- versus 18-gauge, but when you get up to 22-gauge the metal is more prone to denting and vibrating, and less able to handle a garbage disposer. High-gauge drop-ins can be especially thin around the edges, making them ill equipped to support the heavier weight of a quality faucet.
Six-inch-deep basins are cheap to make, but they splash and can't hold much. A 9- or 10-inch sink, on the other hand, holds lots-a big plus when countertop space is limited. Keep in mind that an undermount ends up another inch or so lower, which could strain your back-at which point you might consider investing in a basin rack.