Even asking whether a graph with a given spectrum exists is a tough question. This is witnessed by the open problem of determining whether there exists a graph of girth 5 diameter 2 and order 3250 ...

I think that Sage is the right thing for this task. Example. sage: G = digraphs.Circuit(40) sage: G = G.lexicographic_product(G) sage: G.show() Other graph products and algorithms are available ...

This is just a partial answer mainly answering just $a)$ and $b)$ The number of non-isomorphic graphs is asymptotically equivalent to $$\frac{2^{n \choose 2}}{n!}$$ Almost all graphs are connected (...