Powerful radio galaxies exist as either compact or extended sources, with the extended sources traditionally classified by their radio morphologies as Fanaroff-Riley (FR) type I and II sources. FRI/FRII and compact radio galaxies have also been classified by their optical spectra into two different types: high excitation (HERG; quasar-mode) and low excitation (LERG; jet-mode). We present a catalogue of visual morphologies for a complete sample of >1000 1.4-GHz-selected extended radio sources from the Sloan Digital Sky Survey. We study the environment and host galaxy properties of FRI/FRII and compact sources, classified into HERG/LERG types, in order to separate and distinguish the factors that drive the radio morphological variations from those responsible for the spectral properties. Comparing FRI LERGs with FRII LERGs at fixed stellar mass and radio luminosity, we show that FRIs typically reside in richer environments and are hosted by smaller galaxies with higher mass surface density; this is consistent with extrinsic effects of jet disruption driving the Fanaroff-Riley (FR) dichotomy. Using matched samples of HERGs and LERGs, we show that HERG host galaxies are more frequently star forming, with more evidence for disc-like structure than LERGs, in accordance with currently favoured models of fundamentally different fuelling mechanisms. Comparing FRI/FRII LERGs with compact LERGs, we find the primary difference is that compact objects typically harbour less massive black holes. This suggests that lower mass black holes may be less efficient at launching stable radio jets, or do so for shorter times. Finally, we investigate rarer sub-classes: wide-angle-tailed, head-tail, FR-hybrid and double-double sources.