A total of 717 sources from the Australia Telescope Compact Array (ATCA) catalogue of the Small Magellanic Cloud (SMC) have been classified in Paper II (of this series) by Payne et al. Here, we present a statistical analysis of all 16 confirmed supernova remnants (SNRs) and five new candidate remnants. Included is a detailed discussion of the latter and three other sources that have some SNR characteristics. We have also found a new microquasar candidate (ATCA J005523-721055) in addition to ATCA J004718-723947 reported in Paper II. Source diameter comparisons suggest that SNRs as a group are of similar size in radio, optical and X-ray with surface brightness values in the range of Galactic remnants. Remnant spectral indices, α [defined as Sv α vα, with Sv (flux density) and v (frequency)], have a mean of -0.63 [standard deviation (s.d.) = 0.43] and ROSAT X-ray hardness ratios confirm them to be soft X-ray sources compared to background objects. We could not find any meaningful correlation between SNR surface brightness and diameter; we also discuss the number-diameter relation. A Venn diagram summarizes that most SNRs emit radiation in all three of the radio, optical and X-ray domains. H II region diameter comparisons between radio and optical sources show them to be a very diverse group that defies any simple relationship, preventing any meaningful calculation of flux density or spectral index. To better understand environments containing SNRs, we have scaled Ha images of four SMC regions and subtracted their flux from the ATCA 2.37-GHz radio image. These 2.37 GHz-Ha subtraction (or difference) images reveal some new sources with predominantly non-thermal emission, exposing SNRs confused with H II regions.