For your first question, note that (let $\omega$ be the Hermitian form)

$$\int_M\Delta_c(f) \omega^n=\int_M(\mathrm{tr}_\omega \sqrt{-1}\partial
\overline{\partial}f)\omega^n=n\int_M \sqrt{-1}\partial
\overline{\partial}f \wedge \omega^{n-1}$$

Using Stoke's formula, we deduce that

$$\int_M \partial
\overline{\partial}f \wedge \omega^{n-1}=\int_M \overline{\partial}f \wedge \mathrm{d}\omega^{n-1}=\int_M \overline{\partial}f \wedge \partial \omega^{n-1}=\int_M f \wedge \partial \overline{\partial}  \omega^{n-1}$$

Then
$$
\int_M\Delta_c(f) \omega^n=n \int_M f \wedge \sqrt{-1}\partial \overline{\partial}  \omega^{n-1}
$$

And $\int_M\Delta_c(f) \omega^n=0$ for all $f$ iff $\partial \overline{\partial}  \omega^{n-1}=0$. Such metric is called a $\textbf{Gauduchon metric}$.

The answer to your second question is positive and it was first proved by Gauduchon in "[Le théorème de l'excentricité nulle](https://gallica.bnf.fr/ark:/12148/bpt6k5620896n/f93.item). C. R. Acad. Sci. Paris 285, 387–390 (1977)". 

A more accessible reference is "[The Monge–Ampère equation for non-integrable almost complex structures](https://doi.org/10.4171/JEMS/878)" by Chu, Tosatti and Weinkove. They proved the results in Theorem 2.1, Theorem 2.2 in their paper (for almost Hermitian manifolds):

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/dkLA7.png