Seithr is magical energy that flooded the world after the Black Beast was vanquished by the 6 Heroes (Hakumen, Jubei, etc.) It is toxic in large amounts to humans, so they live way up high.
The Kaka (catgirls) are immune to the negative effects of Seithr so they can survive under the city as long as they want.
The Kaka are clones of one of the powerful weapons used during the Great War (the battle against the Black Beast), so that's why Tao is referred to as an Atreistic Replica. If the Kaka Clan reaches a population exceeding 100, a fail-safe mechanism in their DNA kills them all.
Also, they can asexually reproduce (hot?)
Azure is short for the Azure Grimoire, the most powerful Armagus (technology combined with seithr) in the world, which is wielded by Ragna the Bloodedge (which is referred to as a fake because V-13 and Noel have ones as well, but stronger. When all combined, the universe ends) in his left arm, which was cut off by Hazama when he was a child. Also Jin was with Hazama when they killed Ragna's stepsister and grandmother (his parents were already dead).
Ragna and Jin are step-brothers so gay hentai doujins are NOT incestuous.
Jin IS Hakumen
Noel is V-12
Grimalkin is a racial slur
Rachel is a British, Vampire, Loli who is also God tier and doesn't lose her memory when time loops. She also prevents the end of the world by summoning an encantation that forms a powerful barrier and loops time when Noel's Ars Armagus awakens (Magic Death Ability Thing).
Arakune was once a human archeologist in search of power and wisdom to absorb and control seithr. During his exploits he discovered "The Barrier", a Seithr-filled realm of 'in-between-death-and-life. He was flooded with wisdom and seithr, but was unable to control himself and became a slave to his own desire for power as he transformed into what he is now. He constantly seeks the Azure because it will grant him more power.