Dynamics of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model

New results on the old-standing problem stated in the title of the paper are reported. Emission of radiation by a plane quasi-one-dimensional (Q1D) kink coming across a local defect is investigated in detail. In the same situation, excitation of flexural waves on a "crest" of the Q1D kink is investigated in the long-wave approximation. Next, stable stationary shapes of the kink's "crest" that admit its motion with a constant velocity V are found in the framework of the homogeneous damped dc-driven model. For any V < 1, there is a single v-like profile of the crest. For any "tachyonic" value V > 1, an x-like profile exists (in the absence of the dissipation and the drive terms, it corresponds to an exact solution of the two-dimensional sine-Gordon equation). For a densely packed periodic array of the Q1D kinks, the tachyonic profile is Λ-like.