The question was "Where would magnetism fit in this model?". Kaluza theory says: "It will fit!"

So an advice could be: first get Minkowski-Space and Einstein-Equations done, before including Electromagnetism.

I think, that a light cone should be only one causal knot (!)
(for a photon time stands still and everything is beyond event horizon, but it transports an interaction between generation point and absorption point).

"it is used that if the size of a ball of radius r grows as N=a*r^n then the dimension is n."
That is in general only an indication (only true in the absence of cutvature).
It differs a little bit when there is curvature an r is not small compared to a geodesic curve.

"And also that r^(n+2) terms are associated to curvature."
Where did you read that? But nevertheless, only "associated", only a hint where it goes to.

Curvature is a very complicated thing. Usually you need a metric in a space.
For example take a x,y,z space which is curved an embedded in a 4D space: x²+y²+z²+u²=R².
You can get an impression of the curvature by comparing PI r² with r²=x²+y² with the size of the area included in the circle and you have to do it in the xy, yz and zx plane. For 5D you get even more coordinate planes ...
The total number of algebraically independent components of the curvature tensor
is in N dimensions 1/12 n²(n²-1) n=3--> 6, n=4-->20, n=5-->50
(bottom of https://www.mathpages.com/rr/appendix/appendix.htm)

Here is a classic paper from 1994 that might be of interest with regard to Kaluza-Klein theory. https://arxiv.org/abs/hep-th/9410046

That Kaluza-Klein theory feels very appealing. I am not an expert and I am not quite grasping many aspects of it, but if just by adding a fifth dimension the electromagnetism follows then it is a argument for electromagnetism not being rule specific.

There seem to be several possible compactifications on Kaluza-Klein. And since we are discussing a discrete model we should have more candidates. If we have an structure like the Cartesian product of three large cycles (the three spatial dimensions) and some graph G. For what kind of graphs G we would get the electromagnetism from Kaluza-Klein? I guess that G should have low diameter, so that that 'dimension' is not observable as a spacelike dimension.

Keep in mind, electromagnetism is a special case of a gauge theory. We are not currently trying to find electromagnetism itself, which will likely be rule-specific (although we will search for it in the future). We are now just trying to understand how gauge theories would work in our models in general.