:D I know especially John is very interested in finding an application of this stuff to solve modelling problems for climate change etc, but yes, this is not the content of this work ^_^
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About the choosing which edges they follow: You don't. In standard Petri nets firing is concurrent: If tokens can be used by more than one transition at the same time, they will non-deterministically go one way or another. You can actually refine this situation by extending your formalism, e.g. to timed nets.
I am not an expert of neural nets, but I'd guess they are more similar to signal flow graphs. These are related to Petri nets tho, but in a very deep and complicated way that I have no chance of explaining here right now. Check out the work of Sobocinski, Piedeleu and Zanasi about additive relations if you are interested in this!
The kind of graphical gadget that generates FDHilb (in the sense that the graphical calculus is sound and complete wrt FDHilb) is called ZX calculus (or one of its equivalent variants, such as ZW). It took roughly 10 years to prove that ZX is complete wrt FDHilb! In any case, a string diagram in ZX calculus looks like a hypegraph with extra properties and equations. But you lose the dynamic interpretation of tokens moving in the net, there are no tokens in ZX!
I feel you are one of those people that just hate category theory because they consider it too abstract and useless for any purpose. We get that a lot, from a lot of people. Still, since Grothendieck, categorical methods are absolutely central in modern algebraic geometry and topology, and this centrality is only destined to grow, because CT is the best theory we have to manage emerging complexity in describing systems. In my opinion, the approach of "if it's not immediately useful then it's useless" really will lead you farther and farther away to understand modern developments in applied mathematics.
Applications are not the central focus of this paper. There are about a ton of applied papers out there employing Petri nets in computing, chemistry, epidemiology etc. This paper is not one of them. We just mentioned, in passing, how different flavors of nets have been applied in the last decades. We deem this to be an interesting paper for people working in Petri nets theory, because it systematizes decades of research. I'm pretty sure it will be of little interest for anyone not directly involved in Petri net research. :)
Graphs give you finite state machines in the obvious way: You mark the vertex you are in and walk the arrows.
Hypergraphs give you Petri nets: You mark each vertex as many times as you want and walk the arrows to move marks around. This tells us two things: 1. Petri nets are a calculus of resources. The marking is not telling anymore "what state you are in". A state is an allocation of resources to each vertex in the net. 2. Petri nets are concurrent: you don't have to move stuff around by walking one edge at a time: Two different hyperedges in two different places of the hypergraph can "act at the same time", since the "what state you are in" thing makes no sense anymore.
Anyway, this paper is pretty complicated and for sure there are waaaaay easier places to start. Such as this one: https://arxiv.org/abs/1906.07629
We are working to capture this categorically as well. We are perfectly conscious of how a discrete system won't help you if you have an Avogadro number of things going around. Just give us time. :)
The fact that mathematical foundations don't have applications now does not mean they won't have applications in the future. Gregorio Ricci-Curbastro's work was also considered pretty useless until Einstein decided to build General Relativity over it. If it were for opinions like this one, progress in many field would be incremental at best.