The waves when the water is first entering the area is called numeric dispersion, it's a consequence of discretizing. It can be mitigated somewhat by smoothing the entering wall of water so that there's not a sharp discontinuity.
I don't think so, the waves are actual shock waves which are a solution to the Riemann problem for the shallow water equations because they are nonlinear hyperbolic equations. You can find them even before you start introducing any numerical grid or discretization.
Very interesting, my mistake! I'm more familiar with acoustics and assumed it was the same phenomenon, but it looks like I jumped to a mistaken conclusion.
I agree 100% with this comment -- you can clearly see the energy separating into its component frequencies as the solution evolves. The reason is that different frequencies of the solution propagate at different speeds. Smoothing the discontinuity would indeed reduce the high frequency components, such that the effects wouldn't appear so extreme. The shallow water equations only allow for one shock wave mode as far as I recall, it wouldn't explain why we seem to get increasingly more shocks as the solution evolves.
