Some folks use this as a method of "reverse polarity" protection. It doesn't matter how you connect the + and - DC supply to the input, the output is always between a positive cathode and and negative anode. Disadvantage is you add two forward-biased diode voltage drops in series with your DC power, about a 1.4 volt loss if any significant current flows.
Sure it does. It's a light-emitting DIODE. But why would you want to reverse-bias an LED by applying AC to it? Well, some pairs of LEDs in the same package are designed for this, so forward biasing one yields one color, forward biasing the other yields another color, and if you apply AC it alternately lights both LEDs and yields a mixture of the two colors. However, LEDs are not intended to be used as rectifiers. If the reverse bias voltage is high enough they will break down and destroy themselves.