The spectroscopic parameters and decay channels of the axial-vector tetraquark T-bb;(us) over bar(-) (in what follows T-b;(s) over bar(AV)) are explored using the quantum chromodynamics (QCD) sum rule method. The mass and coupling of this state are calculated using two-point sum rules by taking into account various vacuum condensates, up to 10 dimensions. Our prediction for the mass of this state confirms that it is stable with respect to strong and electromagnetic decays and can dissociate to conventional mesons only via weak transformations. We investigate the dominant semileptonic T-b;(s) over bar(AV) -> Z(b;(s) over bar<)((lv)over bar>)l and nonleptonic T-b;(s) over bar(AV) -> Z(b;(s) over bar)(0) M decays of T-b;(s) over bar(AV). In these processes, Z(b;(s) over bar)(0) is a scalar tetraquark [bc][(us) over bar] built of a color-triplet diquark and an antidiquark, whereas M is one of the vector mesons rho(-), and K*(892), D*(2010)(-) and D-s(*-). To calculate the partial widths of these decays, we use the QCD three-point sum rule approach and evaluate the weak transition form factors G(i)(i = 0,1,2,3), which govern these processes. The full width Gamma(full) = (12.9 +/- 2.1) X 10(-8) MeV and the mean lifetime tau =5.1(-0.71)(+0.99) of the tetraquark TZ(b;(s) over bar)(AV)are computed using the aforementioned weak decays. The obtained information about the parameters of Z(b;(s) over bar)(AV) and Z(b;(s) over bar)(0) is useful for experimental investigations of these double-heavy exotic mesons.