Just to summarize, I answered your question:

it is imperative to make sure that Wolfram model reproduces quantum mechanics to a satisfactory degree

with the reference of the Wolfram Model reproducing ZX calculus (a part of quantum mechanics). The other parts are work-in-progress. Concerning your other question

Representation of the updates in Wolfram Model by linear operators

I think that it is the other way around: the linear operators from quantum mechanics correspond to something in the Wolfram Model, but it is not that something in the Wolfram Model should be expressed as linear operators. I disagree with Jonathan's point of view (I am not saying that he is wrong, I just say that my conceptualization is different from his conceptualization)

we may interpret rewrite relations as being linear operators \hat{A} acting upon this space:

I prefer the other way around: the linear operator should be interpreted in the framework of the Wolfram model. The interpretation of the updates in the Wolfram Model as a linear operator is -in my opinion- a return to the previous paradigm.

From what I can see, ZX calculus is a particular way of doing quantum computing. I am not sure though, what are the conclusions of the ZX paper about the mapping between ZX calculus and Wolfram model. The paper says in the conclusion:

[We] Demonstrated that the diagrammatic rewriting formalism of the ZX-calculus can indeed be embedded and realized within the more general formalism of Wolfram model multiway operator systems, using a novel reformulation of the Wolfram model in terms of double-pushout rewriting systems and adhesive categories.

I do not understand this. ZX calculus is "a graphical language for reasoning about linear maps between qubits", while Wolfram Model is a class of algorithms for updating graphs/strings/etc (with some physical meaning attached to the procedure). What does it even mean, that a language (ZX calculus) is imbedded into the model? Does it mean that "some "words" of the language are realized sometimes in some Wolfram models"? Or does it mean that we can "translate" ZX-calculus language into something made of Wolfram notations? To me, saying "ZX-calculus can be embedded and realized within the more general formalism of Wolfram model", is like saying "calculus can be realized and embedded within a more general formalism of electromagnetism" (in short, nonsense).

I would prefer to use the word "framework" rather than "notation"

What is "framework"? By "translating into Wolfram's notation", I meant just enriching words and notations of Wolfram Models to include definitions from ZX calculus.

I agree that differential and integral calculus can be realized and embedded in the general formalism of electromagnetism

What do you mean by this? Like, the whole phrase is just a non-statement for me... basically, just a collection of words (no intend to insult here, the phrase really doesn't make any sense to me).

The first quote you give is also by Wolfram (not by Gorard) from his technical introduction. Almost all of the phenomena we know comply with quantum mechanics, so it is imperative to make sure that Wolfram model reproduces quantum mechanics to a satisfactory degree. I am talking here about the general structure of quantum mechanics, not some specific examples of particular fields or Schrodinger equation (which also would can be useful).

I am not sure I see how the paper really helps with the questions in the original post. Could you elaborate, please (I am not particularly knowledgeable in ZX calculus or categorical quantum information theory, so would be nice if you could make your point more or less self-contained)?

Regarding the presence of dimensional constants in the theory, you can find some ideas in the technical introduction: Units and Scales.

Unfortunately, I cannot help you with your other questions, because I am still trying to understand the other papers on general relativity (which I think that pedagogically precede the one you mentioned).