tl;dr: If you are talking about VSEPR for an undergraduate level class, core orbitals do not mix. If you are talking about Hartree-Fock theory for a graduate level thesis, core orbitals most certainly do mix.

The answer to this question depends greatly on the level of theory you are looking at.

High-level Quantum Mechanics calculations will typically involve mixing of core orbitals as well as valence orbitals, and give the most realistic picture.

That being said, by making the simplifying assumption that core orbitals do not hybridize (a situation which is almost assuredly untrue), one might make many useful predictions about geometry and reactivity. For example, the entire discipline of Organic Chemistry is based (in part) upon the central assumption that the carbon 1s orbital does not participate meaningfully in hybridization, and yet the discipline still provides many useful tools for discussing structure and reactivity.

tl;dr: If you are talking about VSEPR for an undergraduate level class, core orbitals do not mix. If you are talking about Hartree-Fock theory for a graduate level thesis, core orbitals most certainly do mix.

The answer to this question depends greatly on the level of theory you are looking at.

High-level Quantum Mechanics calculations will typically involve mixing of core orbitals as well as valence orbitals, and give the most realistic picture.

That being said, by making the simplifying assumption that core orbitals do not hybridize (a situation which is almost assuredly untrue), one might make many useful predictions about geometry and reactivity. For example, the entire discipline of Organic Chemistry is based (in part) upon the central assumption that the carbon 1s orbital does not participate meaningfully in hybridization, and yet the discipline still provides many useful tools for discussing structure and reactivity.