One illustrative method disclosed includes selectively forming sacrificial conductive source/drain cap structures on and in contact with first and second source/drain contact structures positioned on opposite sides of a gate of a transistor and removing and replacing the spaced-apart sacrificial conductive source/drain cap structures with first and second separate, laterally spaced-apart insulating source/drain cap structures that are positioned on the first and second source/drain contact structures. The method also includes forming a gate contact opening that extends through a space between the insulating source/drain cap structures and through the gate cap so as to expose a portion of the gate structure and forming a conductive gate contact structure (CB) that is conductively coupled to the gate structure.