A microscopic theoretical framework based on relativistic energy density functionals (REDFs) is applied to studies of shape evolution, excitation spectra, and decay properties of transactinide nuclei. Axially symmetric and triaxial relativistic Hartree-Bogoliubov (RHB) calculations, based on the functional DD-PC1 and with a separable pairing interaction, are performed for the even-even isotopic chains between Fm and Fl. The occurrence of a deformed shell gap at neutron number N = 162 and its role on the stability of nuclei in the region around Z = 108 is investigated. A quadrupole collective Hamiltonian, with parameters determined by self-consistent constrained triaxial RHB calculations, is used to examine low-energy spectra of No, Rf, Sg, Hs, and Ds with neutron number in the interval 158 N 170. In particular, we analyze the isotopic dependence of several observables that characterize the transitions between axially symmetric rotors, γ -soft rotors, and spherical vibrators. An interesting example of a possible occurrence of shape-phase transitions and critical-point phenomena in this mass region is explored.