Is there any relationship between the risk aversion coefficient in an individual's utility function (commonly used in portfolio optimization) and the preference for higher moments such as skewness and kurtosis? In what range, approximately, should they be? Logically, a relationship should exist, as they all somehow represent the investor's attitude towards risk.
There's a huge literature on this topic going back at least 30 years, and I am unfortunately not familiar enough with this literature to give you a great answer to your specific question. However, I will in this answer at least try to point you in some useful directions according to what I've found thus far.
Kurtosis, by the way, seems like it is not addressed directly as part of this literature, perhaps because it is too difficult to measure, or, because it is an even higher moment than skewness, perhaps we should attain a solid understanding of the first 3 moments before we move on to the fourth.
There are at least 3 ways I've come across to answer the question of a relationship between risk aversion and a preference for skewness.
All of these actually try to measure the amount of expected return one gives up in order to get positive skewness, or the amount one gets in exchange for negative skewness. However, combining these estimates with popular estimates of risk aversion measured in the traditional sense could give you a precise answer.
Some references to get you started:
Morningstar recently came out with a piece entitled The Real World is Not Normal: Introducing the new frontier: an alternative to the mean-variance optimizer. It essentially summarizes their views on Mean-CVaR optimization, based on Xiong and Idzorek (2011).
Yes. Check out the Lower Partial Moments literature. In my view the best introduction to this is Narwrocki - A Brief History of Downside Risk Measures. Uryasev established equivalence between CVaR approach and low partial moments. If Markowitz had the tools at the time time, LPM utility functions would be the introductory optimization model as opposed to mean-variance.