Can I pose any bounary data for the wave equation on $[0,infty)$ for given Cauchy data?

I am just going to answer here the problem actually posed in the title.

The one dimensional wave equation can be re-written as $partial_upartial_v phi = 0$, where $u$ and $v$ are the null variables $x + t$ and $x - t$. The initial data then is prescribed on $u+v geq 0, u-v = 0$, while the boundary is $u+v = 0, u-v geq 0$.

So we see that the wave equation implies that the function $psi(x,t) = partial_u phi(x,t)$ solve a transport equation with negative velocity (cf. my second comment above). Thus if your boundary condition is given such that $partial_u phi(x,t)$ is well-determined along the boundary by just the data given there, you will reach an inconsistency. This is one of the ways of seeing why you cannot prescribe simultaneously Dirichlet and Neumann conditions at the same time.

(On the other hand, to make the IBVP well-posed, you need to specify $partial_v phi(x,t)$ along the boundary, since it satisfies a transport equation with positive velocity. That just one of Dirichlet or Neumann conditions suffice follows from the fact that $partial_v phi(0,t)$ can be solved from $partial_u phi(0,t)$ [transported from Cauchy data] and the boundary data.)