1. Q: ?
A: You are right that the boundary codition on the dam should be u(0,y,z,t) = uo (y,z,t) = prescribed . The purpose of the exercise is to express the mathemttical problem in terms of one unknown , \phi, only. The conservation laws and the boundary coditions, etc, are originally expressed in terms of different unknowns such as u, w, and p etc. For mathematical soluton it is more convenient to reduce the problem to one with just one unknown. You are not asked to solve the problem that you formulate.
2. Q: ?
A: Anton (Thomas) points out that in Homework #1, question 2, both integrals should be double, not triple integrals.