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

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

Then $\int_M\Delta_c(f) \omega^n=0$ for all $f$ iff $\overline{\partial} \partial \omega^{n-1}=0$.