A question about holomorphic structure in Atiyah Bott's paper.

Question

I am reading Atiyah & Bott's The Yang-Mills equations over Riemann surfaces. I meet a question about the holomorphic structure.

Let $E\to M$ be a complex vector bundle over a compact Riemann surface. There is a Hermitian metric on $E$.

Then \begin{equation} \mathscr A\cong\mathscr B \end{equation} for \begin{equation} \begin{split} \mathscr A=&\{\text{all smooth hermitian connections}\}\\ \mathscr B=&\{\text{all holomorphic structures on }E\} \end{split} \end{equation}

The identification is described as: given a connection in $\mathscr A$ \begin{equation} d_A\colon\Omega^0(M;E)\to\Omega^1(M;E) \end{equation} then complexify it \begin{equation} d_A\colon\Omega^0_{\mathbb C}(M;E)\to\Omega^1_{\mathbb C}(M;E) \end{equation} and we have \begin{equation} \Omega^1_{\mathbb C}(M;E)=\Omega^{1,0}(M;E)\oplus\Omega^{0,1}(M;E) \end{equation}

Then \begin{equation} d_A=d_A'\oplus d_A'' \end{equation} for \begin{equation} \begin{split} d_A'\colon & \Omega^0_{\mathbb C}(M;E)\to\Omega^{1,0}(M;E)\\ d_A''\colon & \Omega^0_{\mathbb C}(M;E)\to\Omega^{0,1}(M;E) \end{split} \end{equation} and the operator $d_A''$ determines a holomorphic structure of $E$.

Questions:

1. The decomposition $\Omega^1_{\mathbb C}(M;E)=\Omega^{1,0}(M;E)\oplus\Omega^{0,1}(M;E)$ is due to a Hodge star operator. Is there a canonical Hodge star operator here? I know there is another decomposition using the canonical almost complex structure $J$, and I do not understand why not use this.
2. How to recover $d_A$ from $d_A''$? I guess here we need the Hermitian metric.

Thanks in advance.