We report the observation of the rare charm decay $D^0 \to K^-\pi^+e^+e^-$, a search for nine lepton-number-violating and three lepton-flavor-violating neutral charm decays of the type $D^0 \rightarrow h^- h^{'-} \ell^+ \ell^{'+}$, and $D^0 \rightarrow h^- h^{'+} \ell^+ \ell^{'-}$, and a search for seven lepton-number-violating decays of the type $D^{0}\rightarrow X^{0} e^{\pm} \mu^{\mp}$, where $h$ and $h^{\prime}$ represent a $K$ or $\pi$ meson, $\ell$ and $\ell^{\prime}$ an electron or muon, and $X^{0}$ a $\pi^0$, $K^0_S$, $K^{*0}$, $\rho^{0}$, $\phi$, $\omega$, or $\eta$ meson. The results are based on $468$ fb$^{-1}$ of $e^+e^-$ collision data collected at or close to the $\Upsilon(4S)$ resonance with the BaBar detector at the SLAC National Accelerator Laboratory.