First, a little (personal) note on the value of the Physics Project. Science is a way to approach reality, it’s a well-defined method. It consists of theory, predictions, falsifiable experiments and observations. The reduction to ‘theory’ is what also happened to supersymmetry. It’s not because you believe in something that it automatically predicts, explains or unifies the universe. Strictly speaking, the Wolfram Physics project isn’t science and is merely a collection of ideas (with a lot of marketing). Also, by bypassing the peer-review system you automatically put yourself outside the scientific community. Through this I am not saying that the Physics project is not interesting, but let’s not confuse science with an interesting idea. The history of physics (and science as a whole) is full of smart but false (ie. experimentally unproven) ideas.
Regarding your (many) questions. I think you should not take the concept of 'point' too literal and too geometric. For example, in topology open sets consist of 'points' but it's just a name for something. In non-commutative geometric points are inferred from the algebra (of functions). Functions, just like points, are a name for an abstract entity and should also not be taken too literally. In string theory, string leads to some sort of unification but it does not mean that our universe is full of little vibrating strings. In loop quantum gravity you also have a spin network but there as well; the model produces (via deep mathematics) in the end things we can use to make predictions but it does not mean that you literally and geometrically have on the Planck scale a spin foam.
The idea that graphs can lead to fundamental physics is not new. Discrete differential structures, gauge theory and quantum gravity can be defined on graphs (without the Wolfram language). There are many exotic variations on this (topological dynamics, phase transitions on graphs leading to smooth manifolds etc.). Again, it does not mean that on a Planck scale you have points and edges flying around but that as a mathematical model you can reproduce (part of) reality and make predictions. If the predictions clash with reality your model is non-sense. If your model cannot make predictions, it's not physics.
So, I don't think you should see fields, points, manifolds, graphs as something that at the lowest level exist, but abstract entities. Same for many other concepts in physics and mathematics. Wolfram's view of fundamental physics is (presumably) an abstract model of reality, not reality itself.
