re: normit/inverse transformation: Pet theory? This isn't a theory, it's a simple inverse transformation. It's used all the time. George Box showed how to do a 2D version of it in the 1950s. Normit may be a pet name for it, though. It's just a play on words (probit, logit, expit... normit).
As far as describing data, what's wrong with median + IQR for marginal distributions, or some flavor of energy statistic for joint distributions? You will always need to trade off robustness for sensitivity and bias for variance. That's simply a mathematical feature of the universe. There are plenty of ways to take advantage of this to highlight outliers, for example, which often gets you thinking in terms of "hey, this really looks more like a mixture of two completely different distributions" and seeing if that intuition holds up.
The whole point of describing data with summary statistics is that if the assumptions are met this decouples them from the underlying data. If you use the median as an estimator for the center of your distribution and the MAD as an estimator of its scale, you may choose dimensions along which it's not very good at partitioning your observations. If you want a resistant way to describe the expected center of your data, the median and MAD are very useful. Sometimes it's even more useful to plot everything and point out "our results fall into K obvious clusters" each of which will have their own center & spread.
What I'm saying is that there's no silver bullet. Most of the time we take descriptions of the data, see if there are interesting inferences to be drawn, lather, rinse, repeat. "Get a lot more samples" often thrown into the mix. Are there strong clusters in the data? (Usually a plot will show this, whether via projection or in the raw data) Are there continuums that are interesting in relationship to things we care about? (Usually we'll turn around and model their relationship to said thing-we-care-about, conditioned upon a bunch of other items... multivariate regression, which if you're doing it right, will get you plotting the residuals, themselves descriptive of the model fit)
You simply can't do responsible statistical inference without exploring your data to see what's going on. In order to explore complex datasets, there are plenty of techniques, and most all of them demand tradeoffs (see MAD vs. SD or other metrics of "interestingness" for clustering). A number of descriptive statistics ("extremality" for example) rely upon limit behavior of specific distributions and are case-by-case.
I don't think you'll find many silver bullets for either descriptive or inferential statistics. You have to choose your tradeoffs based on what you want to accomplish.