The answer to this question is unknown. It is even unknown if the fraction of graphs on $n$ vertices determined by their spectra converges to 0, or 1, or something between, or even converges at all.

The answer to the first question is unknown. It is even unknown if the fraction of graphs on $n$ vertices determined by their spectra converges to 0, or 1, or something between, or even converges at all.

About the second question, it is not well posed. You have to decide whether you want weighted graphs (in which case you just have symmetric matrices and nearly anything is possible), and you have to decide what is isomorphic for you. You might get some ideas from this paper: C. D. Godsil and B. D. McKay, Constructing cospectral graphs, Aequationes Mathematicae, 25 (1983) 257-268.