Subcritical transition to turbulence in shear flows is characterised by the coexistence of laminar and non-trivial attractors, both of them stable, leading to metastability and space-time intermittency. As a matter of fact, these transitional scenarios are strongly dominated by exact coherent structures ECS, i.e. travelling waves and periodic orbits, emanating from saddle-node bifurcations. While ECS upper branches usually participate in the formation of chaotic attractors, lower branches typically exhibit one unstable eigenvalue and act as a separatrix between the basins of attraction of laminar and non-trivial solutions, thus dictating the amplitude of the disturbances triggering transition. First, we study in the subcritical Taylor-Couette flow the process by which the boundary between the two attractors acquires sensitivity to the initial conditions. Near to the saddle-node, we start by confirming the typical scenario described in the literature, where the separation role of the ECS lower branch persists up to a global bifurcation involving a boundary crisis. At this point, the global bifurcation causes the lower branch to move off the edge of the basin of attraction of the non-trivial state, so that when increasing the Reynolds number, the role of the edge state is then played by a chaotic saddle that appears independently of the travelling waves. Therefore, we analyse how the subsequent equilibria acting as edge states alternate, and demonstrate the importance of unstable solutions within chaotic saddles for understanding the dynamics. Finally, we also analyse the stable chaotic attractor for the same parameter range and show that the dynamics can be very well approximated by a reduction in a one-dimensional discrete map on the Poincare section. We report two different period-doubling cascades at the onset of the chaos, being one of them much cleaner than those previously seen in fluid systems, and allowing us to confirm Feigenbaum's universal theory accurately. Moreover, the probability distribution produced by the chaos and the periodic points embedded in it can be reproduced surprisingly well by the one-dimensional map obtained by interpolation. Remarkably, this provides direct evidence for the existence of infinitely many periodic solutions in fluid chaos via Sharkovsky's theorem.