We present the HOLISMOKES programme on strong gravitational lensing of supernovae (SNe) as a probe of SN physics and cosmology. We investigate the effects of microlensing on early-phase SN Ia spectra using four different SN explosion models. We find that distortions of SN Ia spectra due to microlensing are typically negligible within ten rest-frame days after a SN explosion (< 1% distortion within the 1 sigma spread and less than or similar to 10% distortion within the 2 sigma spread). This shows the great prospects of using lensed SNe Ia to obtain intrinsic early-phase SN spectra for deciphering SN Ia progenitors. As a demonstration of the usefulness of lensed SNe Ia for cosmology, we simulate a sample of mock lensed SN Ia systems that are expected to have accurate and precise time-delay measurements in the era of the Rubin Observatory Legacy Survey of Space and Time (LSST). Adopting realistic yet conservative uncertainties on their time-delay distances and lens angular diameter distances, of 6.6% and 5%, respectively, we find that a sample of 20 lensed SNe Ia would allow us to constrain the Hubble constant (H-0) with 1.3% uncertainty in the flat Lambda CDM cosmology. We find a similar constraint on H-0 in an open Lambda CDM cosmology, while the constraint degrades to 3% in a flat wCDM cosmology. We anticipate lensed SNe to be an independent and powerful probe of SN physics and cosmology in the upcoming LSST era.