Tsunamis are rare, destructive events, whose generation, propagation and coastal impact processes involve several complex physical phenomena. Most tsunami applications, like probabilistic tsunami hazard assessment, make extensive use of large sets of numerical simulations, facing a systematic trade-off between the computational costs and the modelling accuracy. For seismogenic tsunami, the source is often modelled as an instantaneous sea-floor displacement due to the fault static slip distribution, while the propagation in open-sea is computed through a shallow water approximation. Here, through 1D earthquake-tsunami coupled simulations of large M>8 earthquakes in Tohoku-like subduction zone, we tested for which conditions the instantaneous source (IS) and/or the shallow water (SW) approximations can be used to simulate with enough accuracy the whole tsunami evolution. We used as a reference a time-dependent (TD), multi-layer, non-hydrostatic (NH) model whose source features, duration, and size, are based on seismic rupture dynamic simulations with realistic stress drop and rigidity, within a Tohoku-like environment. We showed that slow ruptures, generating slip in shallow part of subduction slabs (e.g. tsunami earthquakes), and very large events, with an along-dip extension comparable with the trench-coast distance (e.g. mega-thrust) require a TD-NH modelling, in particular when the bathymetry close to the coast features sharp depth gradients. Conversely, deeper, higher stress-drop events can be accurately modelled through an IS-SW approximation. We finally showed to what extent inundation depend on bathymetric geometrical features: (i) steeper bathymetries generate larger inundations and (ii) a resonant mechanism emerges with run-up amplifications associated with larger source size on flatter bathymetries.