Omamori Himari/WMG

The whole thing is in the Ghostbusters Universe, set about twenty years afterward.
In the Japanese version of the episode she's introduced in, Lizlet thinks that Yuuto, Himari et al are "Ghost Busters" who have followed her across the world. Obviously Peter, Ray and Egon's little hole-in-the-wall enterprise has expanded into international prestige after the defeat of Vigo the Carpathian, and has set up "Satelite" Ghostbuster stations across the world, or at least to Europe, which Lizlet is explicatly stated to be from.

Yuuto is the great-to-the-nth male-line descendant of Carrot Ironfoundersson.
How the hell else can you explain the way that both of them seem to be able to break up the most dangerous situations by making friends with everyone.
 * That can't be true, since Carrot is a dwarf and Yuuto isn't.

Himari is actually the origonal cat demon.
Himari is actually the origonal cat demon that was once defeated and tamed by Yuuto's ancestor. In one chapter she says something along the lines of "I have always served the Amakawa family" or "I have served the Amakawas for generations." This may be a typho or mistranslation, but if it isn't then I think it means that by the time the current Amakawa takes up Yatsutsuna, she has almost completely regressed to the way she was before being defeated. Then the current scion has to re-"kill" her, which resets her memory and seals her or something similar until a new generation is born. That is why she has no memory of a mother and why she believes that the first ayakashi Yuuto will "kill" will be her.

Yamata no Orochi is gonna show up.
I'm calling it now. This series is way too much of a Japanese mythological creature fest for him to be left out.