I don't know a good answer to your first question (I'd be interested in a good text for that myself), but I can answer the second.
It's easier to explain if we temporarily imagine $\phi$ represents the concentration of some dye made up of little particles suspended in the fluid. The convective term (aka advective term) is transport of $\phi$ due to the fact that the fluid is moving: a single "particle" of $\phi$ will tend move around according to the velocity of the fluid around it. The diffusive term, on the other hand, represents the fact that the dye tends to spread out, regardless of the motion of the fluid, because each particle is undergoing Brownian motion. So if you were moving along at the same velocity as the fluid you would see a small spot of dye tend to become more and more blurred over time.
For quantities like energy and momentum the diffusion happens for a slightly different reason (transfer of the quantity between fluid molecules when they collide) but the principle is the same. The property is transported along with the fluid's bulk velocity (convective term) but also tends to spread out and become blurred of its own accord (diffusive term).
Let's suppose that the boundary is the x-axis. So along the boundary, the stream function is constant. So, $$\frac{\partial^2\psi}{\partial x^2}=0$$ And from the no-slip boundary condition, $$\frac{\partial \psi}{\partial y}=0$$ This can be used to establish the 2nd order finite difference approximation to the value of the vorticity at the boundary:$$\omega=\frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}=\frac{2(\psi(I,1)-\psi(I,0))}{(\Delta y)^2}=\omega(I,0)$$
This is used for the boundary condition on $\omega$.
ADDENDUM
First of all, the stream function is known (and constant) at the solid boundaries (because the solid boundaries are stream lines). So it doesn't have to be solved for. It is determined up to an arbitrary constant, and can thus be taken to be zero at one of the boundaries. At the other boundary, the stream function is equal to the volumetric throughput rate per unit width of channel (which is typically known). So you don't need to solve for the stream function at the solid boundaries.
If the tangential derivative $\partial \psi/\partial x=0$ at all locations along the boundary, it's second partial with respect to x must also be equal to zero. This, of course, all follows from the fact that $\psi$ is constant at the boundary.
To integrate the vorticity equation, you need a boundary condition on the vorticity (or at least a 2nd order finite difference approximation to a boundary condition). Just because $\partial \psi/\partial y=0$ does not mean the the second partial of $\psi$ with respect to y is equal to zero at the boundary; this would imply that the vorticity at the boundary is equal to zero, which we know is not correct.
The variable I in the relationships refers to the I'th x grid point. So, back to the boundary condition on vorticity: We have shown so far that, at the boundary, $$\omega=\frac{\partial^2 \psi}{\partial y^2}$$ subject to the constraint that $\partial \psi/\partial y=0$. If we represent these two conditions in 2nd order finite difference form, we obtain:
$$\omega(I,0)=\frac{\psi(I,1)-2\psi(I,0)+\psi(I,-1)}{(\Delta y)^2}$$and$$\frac{\psi(I,1)-\psi(I,-1)}{2\Delta y}=0$$If we combine these two finite difference equations, we obtain a 2nd order finite difference approximation to the value of the vorticity at the boundary:
$$\omega(I,0)=\frac{2(\psi(I,1)-\psi(I,0))}{(\Delta y)^2}$$
I've successfully used this approach to solving these equations many times.
Best Answer
The speed of sound can be derived from Navier-Stokes equations (cf. Ron Maimon's answer here) or (equivalently) through Bernoulli's equation (cf. Genick Bar-Meir's Fundamentals of compressible Flow Mechanics online text); I've also seen the ideal gas form derived from thermodynamic principles, but cannot find a source at the moment.
So it appears that the speed of sound is independent of the choice of fluid dynamics pictures considered.
Anecdotally, I went through some of the academic-use hydrodynamics codes I have on hand (most of which are for conservative Eulerian hydrodynamics for astrophysical purposes) and the ones that had functions for the speed of sound (some did it in-place, which makes it harder to find using
grep) used the explicit forms (e.g., $c_\text{ideal gas}^2=\gamma p/\rho$). So you should be on safe grounds here.