This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasi-linear coupled system of a parabolic and elliptic PDEs with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness, and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Pruss. Global solutions and controls admitting such are addressed and existence of optimal controls is shown if the temperature gradient is under control. This work is the first of two papers on the three-dimensional thermistor problem.