The South Pole Telescope has discovered one hundred gravitationally lensed, high-redshift, dusty, star-forming galaxies (DSFGs). We present 0.5 resolution 870um Atacama Large Millimeter/submillimeter Array imaging of a sample of 47 DSFGs spanning z=1.9-5.7, and construct gravitational lens models of these sources. Our visibility-based lens modeling incorporates several sources of residual interferometric calibration uncertainty, allowing us to properly account for noise in the observations. At least 70% of the sources are strongly lensed by foreground galaxies (mu_870um > 2), with a median magnification mu_870um = 6.3, extending to mu_870um > 30. We compare the intrinsic size distribution of the strongly lensed sources to a similar number of unlensed DSFGs and find no significant differences in spite of a bias between the magnification and intrinsic source size. This may indicate that the true size distribution of DSFGs is relatively narrow. We use the source sizes to constrain the wavelength at which the dust optical depth is unity and find this wavelength to be correlated with the dust temperature. This correlation leads to discrepancies in dust mass estimates of a factor of 2 compared to estimates using a single value for this wavelength. We investigate the relationship between the [CII] line and the far-infrared luminosity and find that the same correlation between the [CII]L_FIR ratio and Sigma_FIR found for low-redshift star-forming galaxies applies to high-redshift galaxies and extends at least two orders of magnitude higher in Sigma_FIR. This lends further credence to the claim that the compactness of the IR-emitting region is the controlling parameter in establishing the [CII] deficit.
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