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/

I was studying some cloud task, deploy and storage funcitonality in Wolfram Alpha with this project and got these further generated images divided to signatures. I also found there was a similar post made by other person some time ago (but I lost the link to the notebook), so this is an overlapping study, but I just want to get some experience with using Wolfram One and Mathematica before trying it for other projects.

So, I generated double, triple and quad rules with 2 to 5 variables, so at the end I have divisioned signature based on the variable count also. 12223 means rules are generated with one double graph plus two double graphs with three free variables. For example {{{1,2}}->{{2,3},{3,2}}} is such a rule.

Permutation function and reMap function are also a bit simpler now, because I realized that there is no need to use nested list in the beginning, flatten list is ok, which can be reshaped (aShape) to the final rule form at the state of creating Wolfram Model. I also wonder if using numbers (1,2,3) in the rule lists instead of labeled characters (x,y,z) is more robust and uses less memory. Anywa, the new optimized functions are:

reMap[list_] :=
 ReplaceAll[list, 
  Function[aflat, AssociationThread[aflat, Range[Length[aflat]]]][
   DeleteDuplicates[Flatten[list]]]]

 permute[n_, o_] :=
 Permutations[
  Flatten[Sequence @@ ConstantArray[#, n] & /@ Range[o]], {n, n}]

aShape[l_, a_, b_, c_, d_] := 
 {Partition[Take[l, a], c], Partition[Take[l, -b], d]}

Signatures

Note: Not suitable forms means the final state of the wolfram model hypergraph didn't yield conditions that I saw interesting enought to plotted at all.

Sign 1 _ 2 _ 2 _ 2 _ 2

No suitable forms

Sign 1 _ 2 _ 3 _ 2 _ 2

Sign 2 _ 2 _ 3 _ 2 _ 2

No suitable forms

Sign 1 _ 3 _ 2 _ 3 _ 2

Sign 1 _ 3 _ 3 _ 3 _ 2

Sign 2 _ 3 _ 3 _ 3 _ 2

Sign 1 _ 4 _ 1 _ 4 _ 2

No suitable forms

Sign 1 _ 4 _ 2 _ 4 _ 2

Sign 1 _ 2 _ 2 _ 2 _ 3

Sign 1 _ 2 _ 3 _ 2 _ 3

Sign 2 _ 2 _ 3 _ 2 _ 3

Sign 1 _ 3 _ 2 _ 3 _ 3

-Marko