COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses X. Modeling based on high-precision astrometry of a sample of 25 lensed quasars: consequences for ellipticity, shear, and astrometric anomalies

(abridged) Gravitationally lensed quasars can be used as powerful cosmological and astrophysical probes. We can (i) infer the Hubble constant based on the time-delay technique, (ii) unveil substructures along the l.o.s. toward distant galaxies, and (iii) compare the shape and the slope of baryons and dark matter distributions in galaxies. To reach these goals, we need high-accuracy astrometry and morphology measurements of the lens. In this work, we first present new astrometry for 11 lenses with measured time delays. Using MCS deconvolution on NIC2 HST images, we reached an astrometric accuracy of about 1-2.5 mas and an accurate shape measurement of the lens galaxy. Second, we combined these measurements with those of 14 other systems to present new mass models of these lenses. This led to the following results: 1) In 4 double-image quasars, we show that the influence of the lens environment on the time delay can easily be quantified and modeled, hence putting these lenses with high priority for time-delay determination. 2) For quadruple-image quasars, the difficulty often encountered in reproducing the image positions to milli-arcsec accuracy (astrometric anomaly) is overcome by explicitly including the nearest visible galaxy in the model. However, one anomalous system (J1131-1231) does not show any luminous perturber in its vicinity, and three others (WFI2026-4536, WFI2033-4723, and B2045+265) have problematic modeling. These 4 systems are the best candidates for a pertubation by a dark matter substructure. 3) We find a significant correlation between the PA of the light and of the mass distributions in lensing galaxies. In contrast with other studies, we find that the ellipticity of the light and of the mass also correlate well, suggesting that the overall spatial distribution of matter is not very different from the baryon distribution in the inner \sim 5 kpc of lensing galaxies.