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. C. R. Acad. Sci. Paris 285, 387–390 (1977)".
A more accessible reference is "The Monge–Ampère equation for non-integrable almost complex structures" by Chu, Tosatti and Weinkove. They proved the results in Theorem 2.1, Theorem 2.2 in their paper (for almost Hermitian manifolds):
