Using the Monte Carlo (MC) technique, we have analyzed the kinetics of roughening of a clean surface at T>T-R (T-R is the roughening temperature), the kinetics of relaxation from a roughened to a flat surface at T2AB reaction. (i) For a clean surface, the results were obtained by employing the standard restricted-solid-on-solid (RSOS) model with surface diffusion occurring via jumps to nearest-neighbor and next-nearest-neighbor sites. The kinetics of surface roughening at T>T-R and of surface relaxation at TT-R increases with time as w(2) proportional to ln t. (ii) Complementing the RSOS model by terms describing adsorption, we demonstrate that adsorbate-induced surface roughening may occur with significant lowering of T-R, due, respectively, to indirect next-nearest-neighbor adsorbate-substrate repulsion, attractive nearest-neighbor adsorbate-substrate lateral interaction, and/or repulsive adsorbate-adsorbate lateral interaction. In all these cases, the ratio of T-R near adsorbate saturation and for the clean surface, respectively, is given by an analytical expression that is a simple combination of the corresponding interactions. The coverage dependence of T-R in MC simulations is nonlinear and slightly different compared to the mean-field-type prediction. (iii) The effect of adsorbate-induced roughening on a catalytic reaction can be dramatic even at weak roughening, with qualitative and quantitative changes in the kinetics, For example, the absolute conversion rate at maximum rate, and the reactant mixture corresponding to this maximum can be very different on the flat and roughened surface. The behavior of the adsorbate coverages near the points corresponding to the kinetic phase transitions can change character qualitatively.