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This is not a Calabi-Yau if $g\ne 1$ (for any definition of Calabi-Yau). Indeed, there is a degree one map from the symmetric power of the curve to the a torus of dimension $g$. Pull-back of the volume form on the torus to $Sym^gS$ will have zeros at the set where the differential of the map is degenerate, this set is non-empty if $g\ne 1$.
This is not a Calabi-Yau if $g\ne 1$ (for any definition of Calabi-Yau). Indeed, there is a degree one map from the curve to the torus of dimension $g$. Pull-back of the volume form on the torus to $Sym^gS$ will have zeros at the set where the differential of the map is degenerate, this set is non-empty if $g\ne 1$.