Interfacial spin-orbit coupling in Josephson junctions offers an intriguing way to combine anomalous Hall and Josephson physics in a single device. We study theoretically how the superposition of both effects impacts superconductor/ferromagnetic insulator/superconductor junctions' transport properties. Transverse momentum-dependent skew tunneling of Cooper pairs through the spin-active ferromagnetic insulator interface creates sizable transverse Hall supercurrents, to which we refer as anomalous Josephson Hall effect currents. We generalize the Furusaki-Tsukada formula, which got initially established to quantify usual (tunneling) Josephson current flows, to evaluate the transverse current components and demonstrate that their amplitudes are widely adjustable by means of the spin-orbit coupling strengths or the superconducting phase difference across the junction. As a clear spectroscopic fingerprint of Josephson junctions, well-localized subgap bound states form around the interface. By analyzing the spectral properties of these states, we unravel an unambiguous correlation between spin-orbit coupling-induced asymmetries in their energies and the transverse current response, founding the currents' microscopic origin. Moreover, skew tunneling simultaneously acts like a transverse spin filter for spin-triplet Cooper pairs and complements the discussed charge current phenomena by their spin current counterparts. The junctions' universal spin-charge current cross ratios provide valuable possibilities to experimentally detect and characterize interfacial spin-orbit coupling.