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We report the first determination of the relative strong-phase difference between ${D}^{0}\ensuremath{\rightarrow}{K}_{S,L}^{0}{K}^{+}{K}^{\ensuremath{-}}$ and ${\overline{D}}^{0}\ensuremath{\rightarrow}{K}_{S,L}^{0}{K}^{+}{K}^{\ensuremath{-}}$. In addition, we present updated measurements of the relative strong-phase difference between ${D}^{0}\ensuremath{\rightarrow}{K}_{S,L}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${\overline{D}}^{0}\ensuremath{\rightarrow}{K}_{S,L}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$. Both measurements exploit the quantum coherence between a pair of ${D}^{0}$ and ${\overline{D}}^{0}$ mesons produced from $\ensuremath{\psi}(3770)$ decays. The strong-phase differences measured are important for determining the Cabibbo-Kobayashi-Maskawa angle $\ensuremath{\gamma}/{\ensuremath{\phi}}_{3}$ in ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\stackrel{\texttildelow{}}{D}}^{0}$ decays, where ${\stackrel{\texttildelow{}}{D}}^{0}$ is a ${D}^{0}$ or ${\overline{D}}^{0}$ meson decaying to ${K}_{S}^{0}{h}^{+}{h}^{\ensuremath{-}}$ ($h=\ensuremath{\pi}$, $K$), in a manner independent of the model assumed to describe the ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{h}^{+}{h}^{\ensuremath{-}}$ decay. Using our results, the uncertainty in $\ensuremath{\gamma}/{\ensuremath{\phi}}_{3}$ due to the error on the strong-phase difference is expected to be between 1.7\ifmmode^\circ\else\textdegree\fi{} and 3.9\ifmmode^\circ\else\textdegree\fi{} for an analysis using ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\stackrel{\texttildelow{}}{D}}^{0}$, ${\stackrel{\texttildelow{}}{D}}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays, and between 3.2\ifmmode^\circ\else\textdegree\fi{} and 3.9\ifmmode^\circ\else\textdegree\fi{} for an analysis based on ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\stackrel{\texttildelow{}}{D}}^{0}$, ${\stackrel{\texttildelow{}}{D}}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}^{+}{K}^{\ensuremath{-}}$ decays. A measurement is also presented of the $CP$-odd fraction, ${\mathcal{F}}_{\ensuremath{-}}$, of the decay ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}^{+}{K}^{\ensuremath{-}}$ in the region of the $\ensuremath{\phi}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ resonance. We find that in a region within $0.01\text{ }\text{ }{\mathrm{GeV}}^{2}/{c}^{4}$ of the nominal $\ensuremath{\phi}$ mass squared ${\mathcal{F}}_{\ensuremath{-}}&gt;0.91$ at the 90% confidence level.