With the Blue Channel Spectrograph (BCS) on the MMT telescope, we have obtained spectra to the atmospheric cutoff of quasars previously known to show at least one absorption system at z>1.6 with very strong metal lines (candidate metal-strong damped Lya systems; cMSDLAs). The BCS/MMT spectra yield precise estimates of the HI column densities (NHI) of the systems through Voigt profile analysis of their Lya transitions. Nearly all of the cMSDLAs (41/43) satisfy the NHI criterion of DLAs, 10^20.3. As a population, these systems have systematically higher NHI values than DLAs chosen randomly from quasar sightlines. Combining our NHI measurements with previously measured metal column densities, we estimate metallicities for the MSDLAs. These systems have significantly higher values than randomly selected DLAs; at z~2, the MSDLAs show a median metallicity [M/H] ~ -0.67 that is 0.6dex higher than a corresponding control sample. This establishes MSDLAs as having amongst the most metal-rich gas in the high z universe. Our measurements extend the observed correlation between SiII 1526 equivalent width and the gas metallicity to higher values. If interpreted as a mass-metallicity relation, this implies the MSDLAs are the high mass subset of the DLA population. We demonstrate that dust in the MSDLAs reddens their background quasars, with a median shift in the spectral slope of Da = 0.29. Assuming an SMC extinction law, this implies a median reddening E(B-V)=0.025mag and visual extinction A_V=0.076mag. Future studies of MSDLAs offer the opportunity to study the extinction, nucleosynthesis, and kinematics of the most chemically evolved, gas-rich galaxies at high z. [abridged]
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