We briefly summarize current experimental and theoretical results on the two important processes of the low-energy hadron physics involving neutral pions, the rare decay $\pi^0\to e^+e^-$ and the Dalitz decay of $\pi^0$, and provide a new value for the ratio $R={\Gamma(\pi^0\to e^+e^-\gamma(\gamma))}/{\Gamma(\pi^0\to\gamma\gamma)}=11.978(6)\times10^{-3}$, which is by two orders of magnitude more precise than the current PDG average. This value is obtained using the complete set of the next-to-leading-order radiative corrections in the QED sector, and incorporates up-to-date values of the $\pi^0$-transition-form-factor slope. The ratio $R$ translates into the branching ratios of the two main $\pi^0$ decay modes: $\mathcal{B}(\pi^0\to\gamma\gamma)=98.8131(6)\,\%$ and $\mathcal{B}(\pi^0\to e^+e^-\gamma(\gamma))=1.1836(6)\,\%$. Furthermore, we present the complete set of radiative corrections to the Dalitz decays $\eta^{(\prime)}\to\ell^+\ell^-\gamma$ beyond the soft-photon approximation, i.e.\ over the whole range of the Dalitz plot and with no restrictions on the energy of a radiative photon. The corrections inevitably depend on the $\eta^{(\prime)}\to\gamma^*\gamma^{(*)}$ transition form factors. For the singly-virtual transition form factor appearing e.g.\ in the bremsstrahlung correction, recent dispersive calculations are used. For the one-photon-irreducible contribution at the one-loop level (for the doubly-virtual form factor), we use a vector-meson-dominance-inspired model while taking into account the $\eta$-$\eta^{\prime}$ mixing.