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We prove an abstract structure theorem for weighted manifolds supporting a weighted f-Poincaré inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full topological control on two important classes of hypersurfaces of the Euclidean space, namely translators and self-expanders for the mean curvature flow, under either stability or curvature asumptions. As an important intermediate step in order to get our results we get the validity of a Poincaré inequality with respect to the natural weighted measure on any translator and we prove that any end of a translator must have infinite weighted volume. Similar tools can be obtained for properly immersed self-expanders permitting to get topological rigidity under curvature assumptions. [ABSTRACT FROM AUTHOR]