Do core electrons have molecular orbitals?
When valence atomic orbitals start to interact with orbitals on another atom or molecule (e.g. bond formation), the valence electrons respond by forming molecular orbitals with the electrons on the other atom or molecule.  The core electrons also respond to this bonding interaction.  Their energy levels and spatial distributions are perturbed.  The more deeply buried the core electrons, the smaller the perturbation - but the core electrons will also be affected by the bonding interaction.  If one were able to write out the exact wave function to describe the new molecule that was formed, the wave function would include the core electrons.  Their importance would be much less than the valence electrons, but they would be involved in describing the total bonding situation.