It seems we get one from any symmetry of the diagram; are these all of them?
No and no. The $A_n$ diagrams have a diagram symmetry of order $2$ for every $n$, but the induced automorphism of $S_{n+1}$ is inner for every $n$ (it's conjugation by the permutation $k \mapsto n - k$ acting on $\{ 0, 1, 2, \dots n \}$, say). On the other hand, $S_6$ has an exceptional outer automorphism.