What is the probability for a function to be positive-definite?

Just a quick partial answer from me for now as I am in a hurry in my job now. For the determination of whether a polynomial is a "positive-definite" function by your definition, you can easily use Sturm's algorithm. The basic formulation of the Sturm's theorem works for square-free functions, i.e. it fails on a measure zero subset of the set of all the polynomials you are interested in for a given n, hence not influencing the resulting probability.