For (1) you need to understand what the angle of incidence is. That vertical line at E establishes the angle of incidence. In particular, that 60 degrees for the ray FE is its angle of incidence. If you have a ray coming in nearly horizontal (or think of pushing FE down to AE), then the angle of incidence must be nearly 90 degrees, as indicated in the answer. As for the rest, it should be established by Snell's law.
My guess is that (2) is conceptual. I assume that since the ray coming in is nearly horizontal that the refracted ray (i.e., the ray turning inside the medium) at 43 degrees is nearly the critical angle for total internal reflection. Just think about how the critical angle is defined. It is the point at which light would come out horizontal, no? And any adjustment would either allow some light to pass through the surface or be total reflected internally.
From that we should be able to use Snell's law to answer (iii). Is the medium outside the cube supposed to be air? The speed question is also straight-forward if you think about what defines the indices.