We carried out a detailed high-precision study of Ag-like Nd13+ and Sm15+ and In-like Ce9+, Pr10+, Nd11+, Sm13+, and Eu14+ highly charged ions. These ions were identified to be of particular interest for the development of ultra-accurate atomic clocks, the search for variation of the fine-structure constant alpha, and quantum information [Safronova et al., Phys. Rev. Lett. 113, 030801 (2014)]. The relativistic linearized coupled-cluster method was used for Ag-like ion calculations, and a hybrid approach that combines configuration interaction and a variant of the coupled-cluster method was used for the In-like ion calculations. Breit and QED corrections were included. Energies, transition wavelengths, electric-dipole, electric-quadrupole, electric-octupole, magnetic-dipole, magnetic-quadrupole, and magnetic-octupole reduced matrix elements, lifetimes, and sensitivity coefficients to alpha variation were calculated. A detailed study of various contributions was carried out to evaluate uncertainties of the final results. Energies for several similar "reference" ions where the experimental values are available were calculated and compared with experiment for further tests of the accuracy.