Marginally outer trapped surfaces (MOTSs) are the main tool in numerical relativity to infer properties of black holes in simulations of highly dynamical systems. On the one hand, the present work extends previous numerical methods in order to allow tracking of highly distorted horizons in axisymmetry. On the other hand, by applying the new method to a family of initial data as well as to simulations of head-on collisions of black holes, we discover three new phenomena: (i) the merger of MOTSs providing a connected history of the full merger in terms of marginal surfaces without any "jumps", (ii) the formation of self-intersecting MOTSs immediately after the merger, and (iii) a non-monotonicity result for the area of certain smoothly evolving MOTSs.The merger of MOTSs closes a gap in our understanding of binary-black-hole mergers in terms of the quasilocal horizon framework and provides the quasilocal analog of the famous "pair-of-pants" picture of the event horizon of two merging black holes. It allows tracking the evolution of properties such as the area through the highly dynamical regimes from the initially separate to the final common horizon.Through a detailed analysis of geometrical and dynamical properties, we uncover features of the horizons not often considered. In particular, we show why the area increase law for smoothly evolving MOTSs fails to hold in some of the cases analyzed here. Furthermore, we demonstrate a surprisingly direct correspondence of the decay behavior of multipoles and the shear on the outermost horizon with the quasinormal modes of a Schwarzschild black hole.An important role is played by the spectrum of the MOTS stability operator, for which we provide numerical examples of the connection between invertibility of the operator and the existence of a MOTS. Furthermore, we give a prospect of how the full spectrum can become useful for gaining more insight into the merger in absence of symmetries.Finally, a first working generalization of the new numerical algorithm to non-axisymmetric situations is shown, proving the general applicability of the method.