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We present a detailed study of the methodology for correlating `dark sirens'(compact binaries coalescences without electromagnetic counterpart) with galaxycatalogs. We propose several improvements on the current state of the art, andwe apply them to the GWTC-2 catalog of LIGO/Virgo gravitational wave (GW)detections, and the GLADE galaxy catalog, performing a detailed study ofseveral sources of systematic errors that, with the expected increase instatistics, will eventually become the dominant limitation. We provide ameasurement of $H_0$ from dark sirens alone, finding as the best result$H_0=67.3^{+27.6}_{-17.9}\,\,{\rm km}\, {\rm s}^{-1}\, {\rm Mpc}^{-1}$ ($68\%$c.l.) which is, currently, the most stringent constraint obtained using onlydark sirens. Combining dark sirens with the counterpart for GW170817 we find$H_0= 72.2^{+13.9}_{-7.5} \,{\rm km}\, {\rm s}^{-1}\, {\rm Mpc}^{-1}$.  We also study modified GW propagation, which is a smoking gun of dark energyand modifications of gravity at cosmological scales, and we show that currentobservations of dark sirens already start to provide interesting limits. Fromdark sirens alone, our best result for the parameter $\Xi_0$ that measuresdeviations from GR (with $\Xi_0=1$ in GR) is $\Xi_0=2.1^{+3.2}_{-1.2}$. Wefinally discuss limits on modified GW propagation under the tentativeidentification of the flare ZTF19abanrhr as the electromagnetic counterpart ofthe binary black hole coalescence GW190521, in which case our most stringentresult is $\Xi_0=1.8^{+0.9}_{-0.6}$.  We release the publicly available code $\tt{DarkSirensStat}$, which isavailable under open source license at\url{https://github.com/CosmoStatGW/DarkSirensStat}.

Cosmology with LIGO/Virgo dark sirens: Hubble parameter and modified gravitational wave propagation