Is it known if the higher genus (gravitational) Gromov-Witten potential is split in a classical and quantum part like the genus 0 Gromov-Witten potential? If so, Could someone give a reference?
Usually the distinction between classical and quantum is restricted to the genus 0 part: it distinguishes between degree 0 and any degree GW invariants. In particular degree 0, 3-point invariants just recover the ring structure in the cohomology of the target manifold. the quantum cohomology appears as you allow for higher degree curves. Allowing for more marked points gives you a family of Frobenius algebras (the Frobenius manifold).
Higher genera can also be distinguished based on degree restrictions (only degree 0, or any degree). Maybe this is what you were looking for, but I don't see how higer genus, degree 0 invariants can be considered "classical".