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Motivated by a recent experiment on nonlinear tunneling in a suspended carbon nanotube connected to two normal electrodes ͓S. Sapmaz et al., Phys. Rev. Lett. 96, 26801 ͑2006͔͒, we investigate nonequilibrium vibration-mediated sequential tunneling through a molecular quantum dot with two electronic orbitals asymmetrically coupled to two electrodes and strongly interacting with an internal vibrational mode, which is itself weakly coupled to a dissipative phonon bath. For this purpose, we establish rate equations using a generic quantum Langevin equation approach. Based on these equations, we study in detail the current-voltage characteristics and zero-frequency shot noise, paying special attention to the advancement or postponement of the appearance of negative differential conductance and super-Poissonian current noise resulting from electronphonon-coupling induced selective unidirectional cascades of single-electron transitions.