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This thesis consists of three parts in which we explore different topics in pure and applied topology; in Part I, we develop an equivariant version of discrete Morse theory compatible with complexes of groups. In Part II, we link discrete Morse theory to knot projections, and in Part III we apply tools from knotoid theory to the study of knotted proteins. The two overall themes running through the thesis are discrete Morse theory and knot theory. In Part I we construct a version of discrete Morse theory for simplicial complexes endowed with group actions. We characterise the compatibility of an acyclic partial matching on the quotient space with an overlaid complex of groups. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups, and prove that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence. In Part II we obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the sphere. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations. Understanding how knotted proteins fold is a challenging problem in biology. Researchers have proposed several models for their folding pathways, based on theory, simulations and experiments. The geometry of proteins with the same knot type can vary substantially and recent simulations reveal different folding behaviour for deeply and shallow knotted proteins. In Part III, we analyse proteins forming open-ended trefoil knots by introducing a topologically inspired statistical metric that measures their entanglement. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for such proteins. In particular, the folding pathway of shallow knotted carbonic anhydrases involves the creation of a double-looped structure, contrary to what has been observed for other knotted trefoil proteins. We validate this with Molecular Dynamics simulations. By leveraging the geometry and local symmetries of knotted proteins' native states, we provide the first numerical evidence of a double-loop folding mechanism in trefoil proteins.