Is there a theory or theorem of distribution construction so that the construction
satisfies a list of properties?
I'm not really a statistician, but I know there's a collection of results on the
distribution you get when e.g. you add or multiply two variables from given distributions.
These tend to focus on the well-known distributions like Normal. And (from what I have
seen) it really is a collection of case-by-case results, rather than a recipe for creating
a distribution with particular properties... I think this is because statisticians have
found that their well-known distributions are enough for the problems they want to solve.