Nice Work! I answer some of your questions in a fast way since there are many (it is important to ask the question in a precise way so that the reader could easily understand it, examples may help):

Q: how do you decide the initial rule? A: "initial rule?" I guess that you mean the initial hypergraph. It is likely that if such a hypergraph will be found someday, it will be obtained from astronomical data, e.g., the distribution of dark matter. The rule may be found from data about cosmic rays or using particle accelerators. Everything in this answer is speculative.

Q: how do you represent a hypergraph replace the system by formal/axiomatic mathematic equations, is it possible? A: This is a rewriting system, a well-studied mathematical object.

Q: is it possible to collide, fuse, merge different graphs to each other? A: Someone from the Wolfram Physics Project is already exploring the operations between hypergraphs.

Q: is there a way to contribute to the possibly new models repository? A: For contributions, read here https://www.wolframphysics.org/help/

Q: how to categorize Wolfram universe models by their similarity and features? Is there some convenient way of naming them? Cactus, Pillow, and such are nice mnemonics. A: What if a Cactus universe becomes a Pillow universe and then a Cactus again? Before the classification, it is required to know that the hypergraph will converge to some defined shape.

Q: I also would like to ask generally about Theories of Everything, but where should I send my questions? A: Questions about the Wolfram Model are here: https://www.wolframphysics.org/questions/ Questions about other theories of everything could be asked in Eric Weinstein's discord (or in Lex Friedman discord): https://ericweinstein.org/