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

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.

"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)