Let $G$ and $Cay(A,S)$ be strongly regular graphs with the same parameters. Is it true that $G$ is a cayley graph?
no, this is certainly not true. IIRC already on 25 vertices there is a family of 15 non-isomorphic s.r.g.'s with the same parameters, some of them Cayley graphs, some not: see http://www.win.tue.nl/~aeb/graphs/Paulus.html.