I understand the concept of making even and odd extensions of the initial data to satisfy the boundary conditions - using an even extension in the Neumann case and odd extension in the Dirichlet case. This may be a stupid question; for the dirichlet case on the half line the solution is (from the strauss textbook):
$$v(x,t) = \frac{1}{2} \left[ \phi(x+ct) + \phi(x-ct) \right] + \frac{1}{2c} \int_{x-ct}^{x+ct} \psi(y) \, dy \,\,\,\,\,\, \mathrm{for}\,\,\, x>ct$$ $$v(x,t) = \frac{1}{2} \left[ \phi(x+ct) - \phi(ct-x) \right] + \frac{1}{2c} \int_{x-ct}^{ct-x} \psi(y) \, dy \,\,\,\,\,\, \mathrm{for} \,\,\,0<x<ct$$
It's defined by a pair of equations for the two cases $x>ct$ and $0<x<ct$. In the previous section the book solved the case for diffusion on the half line:
$$w(x,t)=\frac{1}{\sqrt{4 \pi kt}} \int_0^{\infty} \left[ e^{-(x-y)^2/4kt} +e^{-(x+y)^2/4kt} \right] \phi(y) \, dy $$
Why can't I just add the two solutions to the wave equation like this? (Although the result looks like it doesn't work, I still don't know what the linear combination actually solves, and how it's related to the original two equations). Is it possible to have a solution that satisfies the boundary in just one equation?
In chapter 9 the book talks about wave equation in 3 dimensions. If I want to solve a neumann problem on the half-space z>0, must it also be given in a set of 2 equations? How do I know when it's okay to add two linear solutions to give another solution? The solution for the 2D case makes sense to me, but this thought has been bothering me; I don't have a great grasp on these concepts.
You don't have two solutions to the wave equation; you have a single piecewise function. $v(x,t)$. The first formula is only valid when $x\ge ct$ (otherwise you'd have negative arguments of $\phi$) the second is only valid when $x\le ct$ (same reason).
As an illustration, consider the simple piecewise function $$ f(x) = \begin{cases} \sqrt{x} \quad &\text{if }x\ge 0; \\ \sqrt{-x} \quad &\text{if }x<0 \end{cases} $$ Formally adding two pieces gives $\sqrt{x}+\sqrt{-x}$, which is pretty much a random string of symbols. It does define a function on $\mathbb{R}$.
The difference between two solutions comes from their general formula:
the solution of diffusion equation is given by an integral over the entire line (due to infinite propagation speed), which results in different pieces of the initial values mixing together instantly.
the solution of wave equation is given by an integral over some part of the line (due to finite propagation speed), which results in different pieces of the initial values coming into play at different moments of time. E.g., the effect of the boundary $x=0$ is felt only when $x<ct$; it's not felt when $x>ct$. Hence, the piecewise solution.
To your general question: you can always add two solutions of a linear homogeneous PDE, thus obtaining another solution. If you actually have two solutions. Having two formulas written on paper may or may not mean that you have two solutions... it rather depends on what the meaning of those formulas.