Yesterday I was browsing through the old discussions on this forum about the stability of 4D atoms (specifically, the existence of the 4D analogue of the hydrogen atom), and decided to google for academic research papers on the subject. One very interesting paper that turned up was Is there a stable hydrogen atom in higher dimensions?. It basically confirms what we concluded, that if we assume an inverse cube law, then there is no stable hydrogen atom.
However, what is very interesting is that the paper pointed out (p.627, third page in the linked pdf) that the reason for this is because we assume Maxwell's equations to hold in higher dimensions. If, instead, we start with general physical properties of, in this case, the desired phenomena, then we can deduce a different set of physical laws that would produce those effects. If we were to formulate 4D electromagnetism in such a way that point charges will have a 1/r potential, for example, then it can be shown that stable hydrogen atoms exist, with analogous (but not the same) properties for electron orbitals.
Section V of this paper (p.631, 7th page in the pdf) derives a modified set of Maxwell's equations that describe higher-dimensional electromagnetism in which stable atoms exist. The force between two charged particles obey an inverse square law, and Gauss's law does not hold (but a modified form does). In even dimensions, the field equations are not differential operator equations, but "pseudo-differential" operator equations, which are non-local. In spite of the non-locality, though, macro-causality continues to hold (see The pseudodifferential operator square root of the Klein-Gordon equation).
In any case, it's clear that electromagnetism cannot be directly generalized to 4D, but a different, modified kind of "electromagnetism" can lead to stable atoms.