We study the processes $e^{+}{e}^{-}\to {{\rm Y}}_J(1D)\eta$ and $e^{+}{e}^{-}\to {{\rm Y}}_J(1D){\pi }^{+}{\pi }^{-}$ at center-of-mass energies $\sqrt{s}$=(10.73 – 11.02) GeV using a $142.5{fb}^{-1}$ data sample, including 122~fb${}^{-1}$ near the ${\rm Y}$(10860) peak ($\sqrt{s}$ = 10.866 GeV), collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. From the peak sample, the products of Born cross section times branching fraction are obtained for $\sigma_{\rm Born}(e^+e^-\to\Upsilon_J(1D)\eta)$ or $\sigma_{\rm Born}(e^+e^-\to\Upsilon_J(1D)\pi^+\pi^-)$ and ${\cal B}(\Upsilon_J(1D)\to\chi_{b1}\gamma)$ or ${\cal B}(\Upsilon_J(1D)\to\chi_{b2}\gamma)$ for each ${{\rm Y}}_J(1D)$ state. The corresponding branching fractions for ${\rm Y}(10860)$ decays are also obtained. The significances of the ${{\rm Y}}_1(1D)$, ${{\rm Y}}_2(1D)$, and ${{\rm Y}}_3(1D)$ signals are 4.8$\sigma$, $>10\sigma$, and 3.0$\sigma$, respectively, including systematic uncertainties. The mass for ${{\rm Y}}_2(1D)$ is measured to be $(10167.0\pm 1.0\pm 0.2)$ MeV/$c^2$, where the first and second uncertainties are statistical and systematic. The mass splittings $\Delta m_{12}=m({{\rm Y}}_2(1D))-m({{\rm Y}}_1(1D))$ and $\Delta m_{23}=m({{\rm Y}}_3(1D))-m({{\rm Y}}_2(1D))$ are $(11.8\pm 1.5\pm 0.4)$ MeV/$c^2$ and $(7.6\pm 2.4\pm 0.6)$ MeV/$c^2$, respectively. We determine the energy dependence of the cross sections for $e^{+}{e}^{-}\to {{\rm Y}}_J(1D)\eta$ and $e^{+}{e}^{-}\to {{\rm Y}}_J(1D){\pi }^{+}{\pi }^{-}$ for the ${{\rm Y}}_1(1D)$, ${{\rm Y}}_2(1D)$, and ${{\rm Y}}_3(1D)$ states, combined.