Hi Harlan,
Thanks for the feedback! Those are good questions, and I can only partially answer them. Let me itemize.
1. Regarding the prediction of the dimension, the first question is how you even define the dimension of a causal relation. It will be emergent, only making sense at large enough scales, and it won't be an integer in general, although it must be very close to 4 at appropriate scales. Fractal dimension is relevant here. There is actually a fair bit of literature on the dimensions of causal sets, but these papers tend to use hypotheses that seem to obscure part of the structure. I have made some progress on this for structures I consider relevant, but it is not yet developed to my satisfaction.
Then, of course, you have to predict it. One of the greatest difficulties with causal theories like causal set theory and some versions of my own ideas is that there are a lot more "obviously nonphysical" universes than physical ones. This is usually described as an "entropy problem," in the sense that nonphysical solutions tend to dominate just like "disordered" solutions dominate in classical statistical thermodynamics. One way around this is to use a Lagrangian approach which (potentially, hopefully!) selects for "physical" behavior by means of an action principle and interference effects. The million-dollar question is then, "what is the 'correct' Lagrangian/action?" Again, I have some ideas about this, but I don't yet know the answer.
2. Regarding antimatter, I can understand it in the context of causal theory only in a very indirect way. In quantum field theory, the necessity for antiparticles "falls out" of the elementary representation theory of the Poincare group, which is the symmetry group of Minkowski space. In causal theory, the Poincare group is replaced with families of refinements of binary relations, and an analogous "representation theory" must be developed. If anyone has done this, I haven't been able to find it, so I am in the beginning stages of doing it myself. There are some aspects of causal theory that make me confident matter-antimatter asymmetry should ultimately be inevitable from this point of view, but I can't explain that at the moment.
3. Regarding the relativity of simultaneity, this is one of the most natural aspects of causal theory. Different frames of reference, rather than merely involving different orderings of spacelike-separated events, ARE different orderings of spacelike-separated events. This prunes away "imaginary geometry" governing what happens, and leaves behind only what actually does happen.
Take care,
Ben