Below is what I have so far. Please let me know if I am correct before I continue.

a) If $v>c$, then $x>ct$. In this region both $\phi(x+ct)$ and $\psi(x-ct)$ are fully defined. We do not need any boundary conditions, which is condition 1.

If $-c<v<c$, then $-ct<x<ct$. Here $\phi$ is fully defined but we need to define $\psi$ for negative arguments. Therefore we need 1 boundary condition, either 2. or 3. (need to work out which one)

If $v<-c$, then $x<-ct$. We need to define both $\phi$ and $\psi$ for negative arguments. We need 2 boundary conditions, so condition 4.