  We present a rigorous microscopic theory of the extrinsic spin Hall effect in
disordered graphene based on a nonperturbative quantum diagrammatic treatment
incorporating skew scattering and anomalous---impurity
concentration-independent---quantum corrections on equal footing. The leading
skew scattering contribution to the spin Hall conductivity is shown to
quantitatively agree with Boltzmann transport theory over a wide range of
parameters. Our self-consistent approach---where all topologically equivalent
noncrossing diagrams are resummed---unveils that the skewness generated by
spin--orbit-active impurities deeply influences the anomalous component of the
spin Hall conductivity, even in the weak scattering regime. This seemingly
counterintuitive result is explained by the rich sublattice structure of
scattering potentials in graphene, for which traditional Gaussian disorder
approximations fail to capture the intricate correlations between skew
scattering and side jumps generated through diffusion. Finally, we assess the
role of quantum interference corrections by evaluating an important subclass of
crossing diagrams recently considered in the context of the anomalous Hall
effect, the $X$ and $\Psi$ diagrams [Ado et al., EPL 111, 37004 (2015)]. We
show that $\Psi$ diagrams---encoding quantum coherent skew scattering---display
a strong Fermi energy dependence, dominating the anomalous spin Hall component
away from the Dirac point. Our findings have direct implications for nonlocal
transport experiments in spin--orbit-coupled graphene systems.
