The questions that you raise in that paper are very related to our research. In particular, we have used the method of moments to show the convergence of a "random linking-number" to the logistic distribution. See [petaluma] by Hass, Linial, Nowik, and me, where we use ideas from work by Mingo and Nica, in [area].
Regarding the first statistics, the number of cyclically increasing triples in a permutation, like 123, 231, and 312, which is asymptotically non-normal: Actually it seems that Gepner's statistics was considered independently by Fisher and Lee. See [angular], where the limiting distribution is described to some extent. It is given as an infinite sum of more standard random variables.
Fisher and Lee give a proof using the theory of U-statistics. I guess that a careful analysis of the moments can yield another derivation. My guess is based on the very close similarity of this distribution to that of another statistics called the writhe that we have derived by the method of moments, in [writhe]. See pages 9-10 and page 16-17.
Regarding the second question, about cyclically increasing triples in balanced 3n-letter words of 1/2/3: This is conjectured to be asymptotically logistic. Again, I strongly believe that it can be derived by the moments method similar to our logistic limit in [petaluma], and probably also by U-statistics methods. However, let me try to convince you by a third line of proof, which is indirect but nice.
First note that the sum over n^3 triples can be replaced by a simpler sum over n^2 pairs, using the tricky identity sign(x-y)sign(y-z)sign(x-z)=(sign(x-y)+sign(y-z)+sign(z-x)), where x,y,z are the locations of 1,2,3 respectively. Now read the word as instructions for a random walk in the plane, where 1,2,3 are three unit vectors whose directions are pairwise 120 degrees apart. This is a closed random walk, and as n grows it becomes similar to a closed 2d Brownian motion. The algebraic area enclosed by such process, the integral of dxdy, was shown by Paul Levy to be asymptotically logistic. It is left to note that the above sum is proportional to that signed area. Indeed, each pair of letters contributes the signed area of some rhombus. This approach is mentioned in [petaluma] in the different setting of four directions.