Why second moment about mean is better at describing spread than the first one? Dispersion is usually used as a measure of inaccuracy of a measurement. It's defined as second moment about mean. Why not define dispersion as cubic root of third moment about mean or as first moment about mean?
 A: Of course you can use other norms, in fact, if you use first moment, linear regression will be robust w.r.t outliers.
Why use higher than first order then? Because, a lot of the time, the utility function is concave, therefore we need a convex function to penalize dispersion/error.
Why second moment then, why not third order? I think see one reason is that second moment usually gives nice analytical closed form solution.
Last but certainly not least, in case you did not notice, we humans observe and understand the world two-dimensionally, the most potent evidence is that we measure our distance in $l^2$ space. 
I can totally imagine in a world where the creatures measure their distance in $l^3$ space study Least Cubic problems instead of Least Square problems, and there is someone posting a question online - "Why do we always measure dispersion using the third moments"?
A: One thing that is nice about the second order moment is that is the smallest order that is differentiable everywhere, and for example, you can minimize parametrized estimators of a variable thanks to calculus.
Also, a lot of our mathematics have been developed in the context of euclidian distance, as it is appears most naturally in our perception of nature. Therefore we have come to be quite proficient with treating distances in terms of L2 spaces.
Another thing is that one likes to state theorems with as much generality as one can, and second order moments exist more often than higher order moments. Therefore, second order moments are more thoroughly studied than third order moments.
