Supercaps are used instead of a small battery because they last a long time. If they installed a standard type of storage battery there, such as lead-acid or lithium-ion, there is a good chance it could decay after some years and not have power available in an emergency, which is the whole point.
4-wheel disc equipped vehicles need a lot more hydraulic pressure to perform a decent stop than cars with drum brakes, so the brake backup capacitor stores energy enough for a few stops with the boost system still enabled even if the 12v electrical system fails. It uses the 12v system to charge itself, and once charged does not consume any significant power. It also allows the complex ABS and hydraulic valve body to continue to function in some capacity.
The parking pawl system used in the Leaf is very similar to what is used in all cars with automatic transmissions, almost identical to the well-proven system used in the Prius, and thus is extremely reliable. The only difference is in the actuation mechanism, which in the Leaf (and in the Prius) is driven by a switched reluctance motor, which is a very reliable type of motor. In most normal automatic transmission cars, the parking pawl is engaged by the mechanical shifter. The parking pawl is strong enough to reliably hold the car and will not "let go". The only possibly significant problem is that if one front wheel were to slip, the car could roll. This is because the parking pawl locks the input shaft to the differential, so only one wheel would need to slip in order to allow the other one to roll. I would consider this to only be a worry if you were parking on a steep hill with a possible traction problem, such as dirt, sand, gravel, ice, etc. In those cases I would also set the parking/emergency brake, but I generally don't bother at any other time. The "roll-click" you sometimes feel when you take your foot of the brake after putting the Leaf into park (or powering it off) is simply the pawl finding the next slot in the toothed cog that allows it to positively lock. Once the pawl locks into the slot, it will not slip out, as it uses a positive locking cam mechanism. The car can roll an inch or so until this happens though, and will make some people uncomfortable but is not dangerous.