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.

*Notation:* $A\hat{\otimes}_{\pi} A$ is the projective tensor product of $A$ with itself. For an $A$-bimodule $M$, the bimodule center is defined by $\mathcal{Z}(A,M) =\{\beta\in M: a\beta=\beta a\hspace{4mm} \forall a\in A \}$. $A^{*}$ is identified with the image of the map (dual of the product map) $\sim:A^{*}\to (A\hat{\otimes}_{\pi} A)^{*}$, which is defined by $\widetilde{f}(x\otimes y) = f(xy)$ on basic tensors (and extended linearly & continuously afterwards).

$A$ 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 Curtis & Loy . 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.
It is also not difficult to show that every $\beta\in \mathcal{Z}(A,(A\hat{\otimes}_{\pi}A)^{*})$ is given by $$\beta(x\otimes y) = f(yx) \hspace{8mm} \forall x,y\in A$$ for some $f\in A^{*}$, and every $\beta\in\mathcal{Z}(A,A^{*})$ is similarly given by $f\in A^{*}$ with the extra property that $af=fa$ for every $a\in A$. Hence, one may identify $\mathcal{Z}(A,(A\hat{\otimes}_{\pi}A)^{*})$ with $A^{*}$, and $\mathcal{Z}(A,A^{*})$ by $C^{\perp}$ in a natural way via $Z(A)$-module isomorphisms. Similarly, $\mathcal{Z}(A,K)$ is isomorphic to a $Z(A)$-submodule of $A^{*}$, call it $B$. Consequently, $$A^{*} = C^{\perp}\oplus B$$ as a direct sum of $Z(A)$ submodules. In particular, $C^{\perp}$ is a complemented subspace of $A^{*}$.

*N.B.:* Amenability is a considerably strong condition compared to the ones given by Q1 & Q2. Thus, it is interesting to see weaker sufficient conditions. On the contrary, Q1/Q2 provides relatively easy to check tests for non-amenability. Thus, it is interesting to see a class of Banach algebras where Q1 or Q2 is not satisfied.

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