For the yellow domain, is it because when t=0 there are infinite possibilities for f(x)? So then there is no unique solution there

Impossible solution in the white domain because the characteristic curve doesn't cross the x-axis so there is no function of x, ie solution doesn't exist

Because the solution u(x, t) is constant on/ determined by the characteristic curves, do we essentially just look at how those curves behave at the initial and boundary conditions?