Using more than 920 fb$^{-1}$ data collected with the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider, we report the first measurement of the $T$-odd moments in the decay $D^{0}\to K^{0}_{S}\pi^{0}\pi^{+}\pi^{-}$. We search for $CP$-violation in decays $D^{0}\to K^{0}_{S}K^{0}_{S}$ and $D^{+}\to\pi^{+}\pi^{0}$. All the results are consistent with no $CP$ violation. We also report the result from the first search for $D^{0}$ decays to invisible final states. No significant signal yield is observed and an upper limit is set on the branching fraction at 90\% confidence level.