The Herschel Multi-tiered Extragalactic Survey (HerMES) has identified large numbers of dusty star-forming galaxies (DSFGs) over a wide range in redshift. A detailed understanding of these DSFGs is hampered by the limited spatial resolution of Herschel. We present 870 μm 0.″45 resolution imaging obtained with the Atacama Large Millimeter/submillimeter Array (ALMA) of a sample of 29 HerMES DSFGs that have far-infrared (FIR) flux densities that lie between the brightest of sources found by Herschel and fainter DSFGs found via ground-based surveys in the submillimeter region. The ALMA imaging reveals that these DSFGs comprise a total of 62 sources (down to the 5σ point-source sensitivity limit in our ALMA sample; σ ≈ 0.2 {mJy}). Optical or near-infrared imaging indicates that 36 of the ALMA sources experience a significant flux boost from gravitational lensing (μ \gt 1.1), but only six are strongly lensed and show multiple images. We introduce and make use of uvmcmcfit, a general-purpose and publicly available Markov chain Monte Carlo visibility-plane analysis tool to analyze the source properties. Combined with our previous work on brighter Herschel sources, the lens models presented here tentatively favor intrinsic number counts for DSFGs with a break near 8 {mJy} at 880 μ {{m}} and a steep fall-off at higher flux densities. Nearly 70% of the Herschel sources break down into multiple ALMA counterparts, consistent with previous research indicating that the multiplicity rate is high in bright sources discovered in single-dish submillimeter or FIR surveys. The ALMA counterparts to our Herschel targets are located significantly closer to each other than ALMA counterparts to sources found in the LABOCA ECDFS Submillimeter Survey. Theoretical models underpredict the excess number of sources with small separations seen in our ALMA sample. The high multiplicity rate and small projected separations between sources seen in our sample argue in favor of interactions and mergers plausibly driving both the prodigious emission from the brightest DSFGs as well as the sharp downturn above {S}880=8 {mJy}. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.