The dissociative sticking of a diatomic molecule on a vibrating surface is a complex many-body process, Here we investigate the dissociation of N-2 on a model Fe substrate with the help of a semiclassical generalized Langevin equation (GLE) scheme, in which the molecule is described by a quantum wave packet, and the substrate idealized by a set of classical Langevin oscillators. In a first part of the paper, cold surfaces at surface temperature T-s=0 K are considered. Here, we investigate (i) the validity of the classical approximation(s) and (ii) errors contained in the widely used single-oscillator approximation, Furthermore, a systematic analysis (iii) of the substrate atom motion and the energy transfer from the molecular (quantum) degrees of freedom to the substrate (classical) modes, and (iv) of effects of increasing "hardness" of the substrate and/or of increasing surface atom masses, is presented. We find that the complex many-body dynamics is more accurately captured by treating many surface oscillators in an approximate fashion, rather than treating a single one as exactly as possible, Ln a second part of the paper, the formalism is generalized to the T-s>0 K case, and effects of finite surface temperatures are studied. It is found that the semiclassical GLE approach correctly accounts for the thermal broadening of the sticking probability vs kinetic energy curve, well-known from experiments on the activated dissociation of diatomic molecules on surfaces.