For your second question, if by probability you mean $$\lim_{n \to \infty} \frac{|S_n|}{|G_n|},$$ where $S_n$ is the set of all possible spectra of simple $n$-vertex graphs, and $G_n$ is the set of isomorphism classes of simple $n$-vertex graphs, then it is conjectured that the above probability is 1. That is, almost all graphs are determined by their spectrum.
See my answer to this question for more information and references.