Aims. We have performed a comprehensive multiwavelength analysis of a sample of 20 starburst galaxies that show the presence of a substantial population of very young massive stars, most of them classified as Wolf-Rayet (WR) galaxies. In this paper, the second of the series, we present the results of the analysis of long-slit intermediate-resolution spectroscopy of star-formation bursts for 16 galaxies of our sample. Methods: We study the spatial localization of the WR stars in each galaxy. We analyze the excitation mechanism and derive the reddening coefficient, physical conditions, and chemical abundances of the ionized gas. We study the kinematics of the ionized gas to check the rotation/turbulence pattern of each system. When possible, tentative estimates of the Keplerian mass of the galaxies are calculated. Results: Aperture effects and the exact positioning of the slit onto the WR-rich bursts seem to play a fundamental role in their detection. We check that the ages of the last star-forming burst estimated using optical spectra agree with those derived from Hα imagery. Our analysis shows that a substantial fraction of the galaxies exhibit evidence of perturbed kinematics. With respect to the results found in individual galaxies, we remark the detection of objects with different metallicity and decoupled kinematics in Haro 15 and Mkn 1199, the detection of tidal streams in IRAS 08208+2816, Tol 9, and perhaps in SBS 1319+579, and the development of a merging process in both SBS 0926+606 A and Tol 1457-262. Conclusions: All of these results - in combination with those obtained in Paper I - reinforce the hypothesis that interactions with or between dwarf objects is a very important mechanism in triggering massive-star formation in starburst galaxies, especially in dwarf ones. It must be highlighted that only deep and very detailed observations - as presented in this paper - can provide clear evidences that these subtle interaction processes are taking place. Based on observations made with NOT (Nordic Optical Telescope), INT (Isaac Newton Telescope) and WHT (William Herschel Telescope) operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway and Sweden (NOT) or the Isaac Newton Group (INT, WHT) in the Spanish Observatorio del Roque de Los Muchachos of the Instituto de Astrofísica de Canarias. Table 1 and Figs. 1-4, 6-10, 12, 13, 15-30, 32-35 and Appendix are only available in electronic form at http://www.aanda.org