The effects of catalyst surface morphology and surface active-sites distribution on diffusion limited reactions are studied. The catalysts structures are deterministic and random Cantor sets (CS) and Devil's staircases (DS). The reaction is an Eley-Rideal process in which a random walker approaches the surface from the bulk and reacts with an active site upon collision with it. Two types of scaling analysis are performed: Fractal time scaling of product formation that provides a global analytical tool of the process; and multifractal analysis with which the complex distribution of reaction probabilities of the active sites are analyzed. Some of the main findings are (a) the fractal time scaling theory that has been applied successfully for connected fractal objects, also describes well the performance of disconnected, dust-type fractal objects such as found in most real catalysts and as modeled by the DS and CS; (b) the effective diffusion coefficient for reactions with disconnected active sets is smaller than for connected sets; we provide a method for calculating it; (c) despite the structural similarity of the DS and CS, they react with different efficiencies. It is shown that this difference can be explained in terms of kv, the prefactor of the neighboring-volume/ yardstick-sized relation; (d) by performing the analyses on objects with various sizes, we provide evidence that corroborates the specific scaling assumptions of the multifractal formalism for the studied structures; and (e) significant differences between the τ(q) and f(α) multifractal spectra of the four objects are found and analyzed.