This thesis aims to contribute to a more thorough understanding of the analysis, numerics and optimization of Maxwell-structured variational inequalities. Motivated by applications that require a certain shielding of electric or magnetic fields, we investigate the evolutionary Maxwell obstacle problem first introduced by Duvaut and Lions and certain variants thereof. To begin with, we analyze the mathematical modeling of the famous eddy current approximation in the Maxwell obstacle problem. Through the usage of an implicit Euler scheme in time and its rigorous convergence analysis, we are able to prove a well-posedness result for the eddy current model. We present uniform a priori estimates that provide information on the quantitative precision of the eddy current approximation. The numerical experiments corresponding to the Maxwell obstacle problem suggest that a combination of the implicit Euler method with a mixed finite element method is too computationally costly. Based on this observation, we introduce an alternative time-discretization by the so called leapfrog stepping. The resulting fully discrete scheme turns out to be indeed way more efficient since it completely eliminates the variational inequality character. We prove the stability and convergence of the fully discrete scheme, which requires us to construct a novel constraint preserving mollification operator for vector fields that admit a weak curl. Thereafter, we study a quasilinear first kind variational inequality with a bilateral differential constraint. We propose a tailored regularization approach by the use of which we examine both the well-posedness and the optimal control of our problem. In particular, invoking curl-projection and cut-off type arguments, we are able to derive a set of necessary optimality conditions for the optimal control problem. Finally, we turn our attention to the construction of an efficient solver for a quasi-variational inequality with applications in superconductivity. Here, we utilize once again a time-discretization by the leapfrog stepping with the aim of eliminating the present quasi-variational character. Exploiting the explicit choice of our nonlinearity, we are able to prove the stability and convergence of the scheme for source and temperature data of merely bounded variation in time. The thesis is concluded with an outlook on the modeling of magnetic levitation phenomena accompanied by numerical experiments.