The physical phenomena of temperature-dependent critical current density and hysteresis losses in high-temperature superconductivity (HTS) lead to a complex multi-physics HTS-system with a two-way nonlinear electromagnetic and thermal coupling. In particular, the coupled problem features a non-smooth character stemming from the governing hyperbolic Maxwell variational inequality for the electromagnetic fields and the governing non-smooth parabolic equation for the temperature distribution. The goal of the reseach project is to develop mathematical and numerical methods for the multi-physics coupled HTS-system with three main aims: First, we shall analyze the existence and uniqueness of solutions by means of the convergence and stability analysis of the implicit Euler method in combination with the Stampacchia method and Fixpoint approaches. Second, we aim at developing and analyzing a semi-smooth Newton algorithm in function spaces and an adaptive mesh refinement strategy on the basis of a residual-type a posteriori error analysis. Finally, based on the achieved mathematical results, we shall investigate the optimal control of the corresponding time-discrete HTS-system with the aim to derive necessary optimality conditions in the form of Pontryagin's maximum principle.