Classical molecular dynamics simulations are used to present a detailed atomistic picture of the instantaneous local structures of water and the structural evolution of stationary and dynamically evolving graphene-water interfaces. The confinement effects are strongly coupled to the nature of the interface, which eventually governs its nanoscopic structural arrangements and interface dynamics. We show that the structure, transport properties, and vibrational densities of states of proximal water molecules are strongly correlated with the nature of the graphene-water interface. We identify and correlate features in vibrational spectra with characteristic structural features observed at the atomic scale for the confined water molecules near a stationary and dynamically evolving hydrophobic surface such as graphene. Our simulations indicate that the local orientation, ordering, and solvation dynamics of interfacial water molecules are a strong function of the graphene slit-width, which is controlled by the nature of the interface (fully flexible vs. static). A monotonic decrease in local ordering with increasing slit-width was observed for the static graphene-water interface, whereas a non-monotonic variation was seen for its fully flexible counterpart. The simulation results offer useful insights into the effect of interfacial dynamics in defining the structure and transport properties at graphene-aqueous media interfaces. Finally these simulations provide a molecular level interpretation of the differential confinement effects arising from the dynamically evolving graphene-water interface.