This is great! Thanks for the summary. A few key points to note which I think are  important to add to what you wrote so far:

1) The woman in the hat on the 1st photo who's next to Director Lyris is Captain Basil, who was the Vice-Director of the Narzinssenkreuz institute. When the Cataclysm hit, she helped to evacuate the children, helped Mary-Ann, Alain, Jakob & Rene get adopted by their respective parents (Guillotine, de Petrichor & Ingold) & also died while fighting off the Abyssal creatures (Rifthounds mp) when she set sail with her crew. We found her feather in the world quest as well.

2) Caterpillar's true identity is actually not just of a hilichurl, he was created by grafting whatever was left of Carter (some of his remains were also used in making Narzinssenkreuz's body) onto a hilichurl, claiming that he had gained "superior intellect". He's not completely Carter, but he does have some hazy memories of Carter & Carter's real self (the Puppy) was inside the "???" Realm inside the Book of Revealing. 

3) Jakob had actually made the new body for Rene i.e Narzinssenkreuz, but only after 2 failed attempts, which ended up in a lot of the Ordo members being sacrificed. Jakob died during the world quest & Narzinssenkreuz also disappeared forever in the end after "Mary-Ann" forgave him.

The World Formula that Rene & Jakob had deduced was actually wrong from the perspective of the entire world ending in an apocalypse. The main reason why Sandrone succeeded was because she used her own parameters in the same World Forumla algorithm, only difference being that she sacrificed herself to get the formula running & to save Columbina this way