One illustrative method disclosed herein includes forming a seed layer above a structure, forming a nucleation layer on the seed layer, forming a plurality of spaced-apart, vertically oriented alloy structures that are comprised of materials from the seed layer and the nucleation layer, forming a sacrificial material layer above the nucleation layer and around the alloy structures, performing an etching process to remove the alloy structures and portions of the seed layer so as to thereby define a plurality of openings, forming an initial masking structure in each of the openings, performing an etching process to remove the sacrificial material layer and the nucleation layer so as to thereby expose the structure and define a masking layer comprised of the initial masking structures, and performing at least one process operation on the structure through the masking layer.