Optimal upper bound on the number of non-isomorphic graphs with certain parameter
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 (in fact $k$-connected for any constant $k$) hence this answer your first question.
This is an open problem. If we restrict ourselves to trees then it is well known that almost all trees are cospectral hence that for every tree $T$ there exist another non-isomorphic tree $T'$ with equal adjacency spectrum. I believe there is no other result generalizing this one.  There is plenty of literature on this subject, check on Google for "cospectral graphs". 


It is conjectured that almost all graphs are determined by their spectrum but no real evidence for this fact is known. The conjecture is of course motivated by the isomorphism problem. 

The number (and proportion) of cospectral graphs on $n=12$ vertices has been computed recently and can be found here.