We report on the properties of radio-selected galaxies within 30 very rich Abell clusters with z less than or similar to 0.25. The radio, optical, and X-ray data for these clusters were presented in Paper I. These radio data sample the ultra-faint (L-1.4 greater than or equal to 2 x 10(22) W Hz(-1)) radio galaxy population with M-R less than or equal to -21 using the well-known FIR-radio correlation to link the radio with ongoing star formation within individual cluster galaxies. Spectroscopic redshifts exist for similar to96% of the optical identifications. These radio-selected galaxies reveal the active galaxy population (starbursts and active galactic nuclei [AGNs]) within these rich cluster environments that can be identified regardless of their level of dust obscuration. These new radio data provide the largest sample to date of low-luminosity radio galaxies within rich cluster environments, allowing an unbiased search for dusty starbursting galaxies. For all clusters in our sample, we are sensitive to star formation rates (M greater than or equal to 5 M-circle dot) greater than or similar to5 M-circle dot yr(-1). We have found that the excess number of low-luminosity starburst radio-selected galaxies (SBRGs) found previously by Owen et al. in Abell 2125 is not indicative of other rich clusters in our sample. The average fraction of SBRGs is (f(SBRG) ) = 0.022 +/- 0.003. The A2125 fraction is f(SBRG) = 0.09 +/- 0.03, which is significantly different from the sample average at a greater than 99.99% confidence level. Both A1278 and A1689 are slightly different from the rest of the sample at similar to90% confidence level. The bimodal structure of both the X-ray brightness distribution and optical adaptively smoothed images of A1278 and A2125 suggests that ongoing cluster-cluster mergers may be enhancing this SBRG population. The A1689 excess low-luminosity ( and high-luminosity) radio galaxy population may be due to interaction with the intracluster medium. The mid-infrared ISOCAM results for A1689's radio galaxy population suggest that the radio emission for both low- and high-luminosity radio galaxies is AGN in origin except for one radio galaxy. There is a significant spatial distribution difference between the low- and high-luminosity (HLRG) radio-selected populations. The SBRGs have a core radius of 0.40 +/- 0.08 Mpc, which is greater than 3 times larger than the HLRG core radius. In addition, 48% of the SBRGs have colors that are bluer than a typical Sab galaxy, compared with 4% for the HLRGs. The average absolute magnitude for the SBRGs is (M-R) = 21.93 +/- 0.05, while for the HLRGs it is (M-R) = 22.33 +/- 0.07, indicating that the SBRGs are less optically luminous than their HLRG counterparts. The HLRGs seem to be a subclass of the cluster's massive red elliptical population, while the SBRGs have a projected radial distribution more like the blue spiral population. Our results indicate that most of the SBRGs are probably gas-rich disk galaxies undergoing greater than or similar to5 M-circle dot yr(-1) of star formation.