Why are the polarized caps backwards?
The capacitors that are "facing the wrong way" are not actually facing the wrong way. The white "plate" indicates the side with the higher potential voltage. If you look at C4, the white plate is on the +Vs side and the black plate is on the ground side. If you look at C6, on the other hand, the white plate is on the ground side, but that is because the black plate is connected to a negative voltage (-Vs). The ground is at a higher voltage than the -Vs supply, so the white plate connects to the higher potential.

I do not see any reason why you couldn't use non-polarized capacitors, but usually the higher capacitances are most common in polarized packages (i.e. electrolytics or tantalums). I imagine all of the polarized capacitors in this circuit are either electrolytic or tantalum, and have a higher capacitance to act as storage and filtering.