A measurement of the cross section for $K^0_S$ pair production in single-tag two-photon collisions, $\gamma^*\gamma\rightarrow K^0_SK^0_S$, for $Q^2$ up to 30 GeV$^2$ is reported. Here $Q^2$ is the negative of the invariant mass squared of the tagged photon. For the first time, the transition form factor of the $f^\prime_2(1525)$ meson is measured separately for the helicity$-$0, $-$1, and $-$2 components and compared with theoretical calculations. The $\gamma^*\gamma$ partial decay widths of the $\chi_{c0}$ and $\chi_{c2}$ charmonia are measured as a function of $Q^2$. The measurements of $\gamma\gamma\rightarrow\eta_{c}(1S), \eta_{c}(2S)\rightarrow\eta'\pi^{+}\pi^{-}$ with $\eta'$ decays to $\gamma\rho$ and $\eta\pi^{+}\pi^{-}$ are reported as well. First observation of $\eta_{c}(2S)\rightarrow\eta'\pi^{+}\pi^{-}$ with a significance 5.5$\sigma$ including systematic error is obtained. The products of the two-photon decay width and branching fraction of decays to $\eta'\pi^{+}\pi^{-}$ are determined for the $\eta_{c}(1S)$ and $\eta_{c}(2S)$, respectively. The cross section for $\gamma\gamma\rightarrow\eta'\pi^{+}\pi^{-}$ and $\eta'f_{2}(1270)$) are measured for the first time. These results for the $K^0_SK^0_S$ ($\eta^\prime\pi^+\pi^-$) production are based on the data sample of 759 fb$^{-1}$ (941 fb$^{-1}$) collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider.