i worked on DEM errors assessing the effects on skewness of DEM-derived parameters (Curvature). https://www.researchgate.net/publication/233968571_Variations_in_multiscale_curvature_distribution_and_signatures_of_LiDAR_DTM_errors?ev=prf_pub
About the effects of errors on the other statistics, here another work where we analyzed errors on DEM from structure from motion
more generally on DEM errors, you could also read the works by J Oksanen on DEM errors...
i would also add that for errors in a DEM you should also consider the spatial correlation and distribution of the errors, rather than just the statistics.
Article Variations in multiscale curvature distribution and signatur...
Article Bank erosion in agricultural drainage networks: new challeng...
The FQ, SQ (median), and TQ are positional measures to divide the arranged data into four equal parts. However, Kurtosis and skewness are measures of Normal distribution.
So, I guess, the positional measures say (FQ) is a point, below which 25% of area falls.
May be you will be interested to get many other variables from the DEM, and wishes to compare between the four areas (classified based on Quartile). That is why, you wish to see the distribution by K and S.