We study domain growth and interfacial melting in a two-dimensional lattice-gas model by means of Monte Carlo simulation, using the grand-canonical ensemble. The model exhibits two classes of "solid" domains with superstructure unit cells of different orientation with respect to the underlying square lattice. Interfacial melting (grain-boundary melting) is only observed for grain-boundaries between domains of different orientation. For a constrained configuration of two domains of different orientation ("bicrystal"), separated by a single interface in the [1 1] direction, we find a logarithmic divergence of the width of the disordered wetting layer in accordance with predictions from mean-field theory. Domain growth in an unconstrained and initially disordered system, following a (down-)quench to temperatures well below the bulk melting temperature T_m, is found to be described by the the Lifshitz-Allen-Cahn growth law with exponent n = 1/2. With increasing quench temperature, a transition from non-activated continuous ordering to an activated nucleation-and-growth mechanism is observed. For quenches to temperatures only slightly below T_m, domain growth turns out to be effectively suppressed. When not yet equilibrated multi-domain configurations are up-quenched to and annealed at temperatures slightly below T_m, the further evolution of the system depends on the morphology of the domain pattern: an equilibrium mono-domain configuration is eventually obtained only if domains of the same orientation were initially present. For initial configurations with domains of both types of orientation, however, and depending on the actual domain sizes, interfacial melting may again drive the system towards a completely disordered, long-living metastable state.