  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]
