Light slows down while passing through, say, water. After passing through the water, does the light speed back up to the "speed of light"? If so, how and why?
Classical electromagnetic waves, which is what light is, emerge as a confluence of an enormous number of photons, quantum mechanical entities. Light does change velocity through the medium, the photons do not, they always travel at c.
What happens in materials the build up of light by photons changes, because photons undergo elastic and inelastic scatterings and the built up wave has a smaller velocity than c, while the individual photons still have velocity c, due to these elementary quantum mechanical interactions. This appears in the index of refraction given in the other answer.
When there is no matter intervening the photons and the beam they build up have the same velocity c.
yes. passed each interface, the new refraction index tells the new celerity (and wavelength change). See here Huygens formulation and graphical derivations.
Without going into the thick of it: light doesn't really slow down inside a transmissive material, so it doesn't speed up upon exiting either.
What it does is interact with the particles of the material, and it is this interactions that slows the effective time of travel.