We use dense redshift surveys to explore the properties of galaxy clusters selected from the redMaPPer catalog. Our new survey, Hectospec Cluster Survey of red-sequence-selected clusters (HeCS-red), includes 10589 new or remeasured redshifts from MMT/Hectospec observations of 27 redMaPPer clusters at redshifts z=0.08-0.25 with large estimated richnesses (richness estimate {lambda}>64). Our spectra confirm that each candidate cluster corresponds to a large overdensity in redshift space. The redMaPPer photometric redshifts have a slight bias toward higher redshifts. We measure the relation between velocity dispersion {sigma}p and {lambda}. The relation shows intrinsic scatter of 24% (and a range of 2x) in velocity dispersion at fixed richness. We extend our analysis to HeCS-red-ext, a sample that includes several clusters selected by X-ray flux or Sunyaev-Zeldovich signal. The heterogeneous sample of 121 clusters in HeCS-red-ext shows similar intrinsic scatter in the {sigma}p-{lambda} relation as the HeCS-red sample, but the range of {sigma}p at fixed richness increases to 3x. We evaluate the membership probability estimates Pmem for individual galaxies provided by redMaPPer. The spectroscopic membership fraction is larger than Pmem for 0.05<=P_mem_<=0.7; conversely, the spectroscopic membership fraction is smaller than P_mem_ at P_mem_>=0.8. We compare spectroscopic richness estimates with redMaPPer richness estimates and find good agreement on average, but a range of a factor of two in spectroscopic richness at fixed redMaPPer richness. Overall, within the high-richness and low-redshift cut of our sample, spectroscopically estimated parameters such as velocity dispersion correlate well with photometric richness estimates, although the relations contain substantial scatter.