Isolation Forest Score Function Theory
I'm not certain on your first question; the paper does seem poorly written in this part.  They seem to be arguing that taking just $E(h(x))/n$ or $E(h(x))/\log n$ are inappropriate.  The former tends to zero regardless of $x$, but the latter should be reasonable.  Indeed, since $c(n)\sim 2\log n$, it's not far off from what they end up doing.  (Exponentiation pushes the scores into the range $[0,1]$ which is nice, and using the more exact $c(n)$ calibrates things better for "(c) if all the instances return $s\approx 0.5$, then the entire sample does not really have any distinct anomaly.")

The rest of the confusion seems to be stemming from what set we're averaging over.  In the isolation forest paper, the averaging is of the depths of (the leaf containing) $x$ over multiple trees in the isolation forest.  In the other reference, the average is of the height of all binary trees on $n$ nodes.