We establish the generator polynomials of the Euclidean and Hermitian duals of separable constacyclic codes over $\mathcal{S} = \mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)$, with $q$ an odd prime power and $v^2=v$, and we derive the generator polynomials of their Gray images, respectively. The generator polynomials of the Euclidean hulls and Hermitian hulls of separable constacyclic codes over $\mathcal{S}$ and their Gray images are presented, respectively. Furthermore, we provide the generator polynomials of the Euclidean sums and Hermitian sums of separable constacyclic codes and their Gray images, respectively. Finally, we propose two methods to yield quantum error-correcting codes (QECCs) from the hulls and sums of separable constacyclic codes over $\mathcal{S}$, and generate new QECCs that outperform the existing ones in terms of parameters.

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