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We systematically determine ground-state and saddle-point shapes and masses for 1305 heavy and superheavy nuclei with $Z=98-126$ and $N=134-192$, including odd-$A$ and odd-odd systems. From these, we derive static fission barrier heights, one- and two-nucleon separation energies, and $Q_α$ values for g.s. to g.s transitions. Our study is performed within the microscopic-macroscopic method with the deformed Woods-Saxon single-particle potential and the Yukawa-plus-exponential macroscopic energy taken as the smooth part. We use parameters of the model that were fitted previously to masses of even-even heavy nuclei. For systems with odd numbers of protons, neutrons, or both, we use a standard BCS method with blocking. Ground-state shapes and energies are found by the minimization over seven axially-symmetric deformations. A search for saddle-points was performed by using the "imaginary water flow" method in three consecutive stages, using five- (for nonaxial shapes) and seven-dimensional (for reflection-asymmetric shapes) deformation spaces. The results are collected in two main tables. Calculated ground-state mass excess, nucleon separation- and $Q_α$ energies, total, macroscopic(normalized to the macroscopic energy at the spherical shape) and shell corrections energies, and deformations are given for each nucleus in \mbox{Table 1}. \mbox{Table 2} contains calculated properties of the saddle-point configurations and the fission barrier heights. In \mbox{Tables 3-7}, are given calculated ground-state, inner and outer saddle-point and superdeformed secondary minima characteristics for 75 actinide nuclei, from Ac to Cf, for which experimental estimates of fission barrier heights are known. These results are an additional test of our model.