The purpose of this note is not to answer my own question, but to share a sufficient condition with everyone: If $A$ is amenable, then Q1 & Q2 are answered affirmatively.
In fact, $A$ is amenable if and only if $(A\hat{\otimes}_{\pi}A)^{*} = A^{*} \oplus K$ as a direct sum of $A$-bimodules for some sub-$A$-bimodule $K$ of $(A\hat{\otimes}_{\pi}A)^{*}$, see https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-40.1.89. From this, it is not difficult to show that $$\mathcal{Z}(A,(A\hat{\otimes}_{\pi}A)^{*}) = \mathcal{Z}(A,A^{*}) \oplus\mathcal{Z}(A,K)$$ as a direct sum of $Z(A)$-modules.