Canonical Separation of variables Do the functions of the form $\psi(x)\phi(y)$ span $L^2(\mathbf{R}^6)$? Insert proper grammar here.

They do indeed. More generally, $L^p(P)\otimes L^p(Q) = L^p(P\times Q)$. A monograph on such seperated / tensor product representations is Hackbush's "Tensor Spaces and Numerical Tensor Calculus" (2012)