Fey creatures intentionally Disneyfied themselves to be seen as less of a threat to humans.A couple hundred years ago, the Fey had been happily doing their own thing and messing with humans for centuries, but at one point they realised that the humans keep building themselves sharper swords, tougher armour and big sticks that go bang. Seeing at least some of the devastating weapons to follow, they decided that they'd best avoid retaliation for their 'fun' and being wiped out by changing their image to playful, helpful and cute creatures from Fairyland, and such helpful spirits as the Tooth Fairy and (possibly) Santa Claus. Seeing things revert somewhat to their old image is probably confusing them a bit, but also at least letting them get a bit nostalgic and go punk rock.
The Fair Folk (as we think of them) didn't let themselves get Disneyfied, they just went underground and let other races do the PR work for them.Santa's elves? German brownies. Nobody finds out what happens when they don't get paid because Nicholas/Chris isn't that stupid or cruel, and they have no need or want to go to the television stations of the people who make films about elf strikes being unionized or not effective enough. The Tooth Fairy is all that remains of an anti-legend for beings that steal teeth from the garbage or the ground for sympathetic magic, so the children needed to dispose of them in a way that would make it difficult for the fey to get at them; More recent times have parents outright telling the kids to leave the teeth for the "Tooth Fairy" because they don't ken the power of words and intent. Most of the Fair Folk live underground these days, either literally in the case of more feral earthbound races and those who find abandoned human subways interesting places to set up court or metaphorically in the case of those who stop leaving the pet flap open on their hills to keep the squishy, easily-confused, two-legged toys from getting in as often.
The Green Movement, if they've noticed, is simply telling them that all they have to do is wait for us to willingly give up iron... of all kinds. Since we've mostly forgotten about the old ways, it's only a matter of time...
The Fair Folk are harmed by cold iron because of its ability to affect magnetic fields.Perhaps in a similar way to a shark, Fay can naturally detect magnetic fields (possibly an explanation for how they can distort our minds (they're using magnetism to screw around with our brains). Contact with cold iron cancels out this ability, making Fay extremely vulnerable.
Cats are a kind of Fair Folk.Let's see... inscrutable? Check. Can entice humans with their cute appearance? Check. Sees us as more their pets than the other way around? Check. It's not for nothing that Terry Pratchett gave us the page quote for Cats Are Mean in a book about the Fair Folk.
The Fair Folk have glamoured themselves in the process of Disneyfication in order to remove human protections against them.Before Grimm, there was a solid body of folk lore about the Fair Folk and how to keep yourself safe. (e.g. stay on the path, don't accept any deals from strangers, cold iron and salt can protect you from their powers and so on...) However Grimm and later Disney diluted and in many cases directly reversed the established knowledge about them. To create a society which believes in the power of wishes, trusts people based on beauty, that if you're a woman that all you need to do is wait and do nothing until a hero rescues you and teaches that running off into supernatural lands will solve your problems. Then when we are thoroughly indoctrinated, they will come back and feast on us once more.