We investigate the prompt emission and the afterglow properties of short-duration gamma-ray burst (sGRB) 130603B and another eight sGRB events during 2012-2015, observed by several multiwavelength facilities including the Gran Canarias Telescope 10.4 m telescope. Prompt emission high energy data of the events were obtained by INTEGRAL-SPI-ACS, Swift-BAT, and Fermi-GBM satellites. The prompt emission data by INTEGRAL in the energy range of 0.1-10 MeV for sGRB 130603B, sGRB 140606A, sGRB 140930B, sGRB 141212A, and sGRB 151228A do not show any signature of the extended emission or precursor activity and their spectral and temporal properties are similar to those seen in case of other short bursts. For sGRB 130603B, our new afterglow photometric data constrain the pre-jet-break temporal decay due to denser temporal coverage. For sGRB 130603B, the afterglow light curve, containing both our new and previously published photometric data is broadly consistent with the ISM afterglow model. Modeling of the host galaxies of sGRB 130603B and sGRB 141212A using the LePHARE software supports a scenario in which the environment of the burst is undergoing moderate star formation activity. From the inclusion of our late-time data for eight other sGRBs we are able to: place tight constraints on the non-detection of the afterglow, host galaxy, or any underlying `kilonova’ emission. Our late-time afterglow observations of the sGRB 170817A/GW170817 are also discussed and compared with the sub-set of sGRBs.