Common fixed points of representations of different categories of topological and analytic objects have been a pivotal area of evolving interest in the studies of fixed-point theory and harmonic analysis for several reasons. In this text, we consider certain families of left/right coset, double coset, and orbit spaces arising from the category of locally compact groups. We solely investigate their actions on compact subsets of general locally convex spaces, as well as on certain Banach spaces. In particular, we use some recent developments in abstract harmonic analysis regarding the theory of Semihypergroups to provide an overview of several characterizations for the existence of common fixed points of such actions in terms of amenability of the underlying spaces.