The present paper discusses the results from an analysis of the images presented in Paper I (astro-ph/0207373) supplemented with new spectroscopic data obtained at Keck. All but one object in the 1-Jy sample show signs of a strong tidal interaction/merger. Multiple mergers involving more than two galaxies are seen in no more than 5 of the 118 (< 5%) systems. None of the 1-Jy sources is in the first-approach stage of the interaction, and most (56%) of them harbor a single disturbed nucleus and are therefore in the later stages of a merger. Seyfert galaxies (especially those of type 1), warm ULIGs (f_{25}/f_{60} > 0.2) and the more luminous systems (> 10^{12.5} L_sun) all show a strong tendency to be advanced mergers with a single nucleus. An analysis of the surface brightness profiles of the host galaxies in single-nucleus sources reveals that about 73% of the R and K surface brightness profiles are fit adequately by an elliptical-like R^{1/4}-law. These elliptical-like 1-Jy systems have luminosities, half-light radii, and R-band axial ratio distribution that are similar to those of normal (inactive) intermediate-luminosity ellipticals and follow with some scatter the same mu_e - r_e relation. These elliptical-like hosts are most common among merger remnants with Seyfert 1 nuclei (83%), Seyfert 2 optical characteristics (69%) or mid-infrared (ISO) AGN signatures (80%). In general, the results from the present study are consistent with the merger-driven evolutionary sequence ``cool ULIGs --> warm ULIGs --> quasars, although there are many exceptions. (abridged)
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