An important aspect of solving the long-standing question as to what triggers various types of active galactic nuclei (AGN) involves a thorough understanding of the overall properties and formation history of their host galaxies. This is the second in a series of papers that systematically study the large-scale properties of cold neutral hydrogen (H I) gas in nearby radio galaxies. The main goal is to investigate the importance of gas-rich galaxy mergers and interactions among radio-loud AGN. In this paper, we present results of a complete sample of classical low-power radio galaxies. We find that extended Fanaroff & Riley type-I radio sources are generally not associated with gas-rich galaxy mergers or ongoing violent interactions, but occ...