Foul seems to be something we picked up by being near the creator of black magic during the 60-70 questline. As to what it is the requirement for casting foul is to have a polyglot which we get from building of aether from enochian. Now the English definition of a Polyglot is someone who learns multiple languages as an avocation, so by learning Foul I assume its our character learning a new type of magic, akin to us learning a new language. Now as to the what type of magic Foul is I like to think it's us opening a temporary tear from Eorzea into the void, using the built up aether from enochian, to corrupt or foul the aether of living beings in the surrounding area. This is just a guess though, Foul could be something else entirely, but learning Void magic, or another type of magic altogether, seems to make sense as our characters become Polyglots of magic.
As for our AF gear its probably a gift from Shatatto herself, but knowing her feelings about large gaudy hats, the headpiece we receive is kind of ironic. Or perhaps the headpiece was made that way to make fun of her a bit, who knows.