In recent years, numerical methods that preserve physical properties and mathematical structures have become a key focus in the field of magnetohydrodynamics (MHD) simulation. The MHD system encompasses several important mathematical structures and physical properties. Designing numerical methods that accurately preserve these structures and properties has become a critical research topic in MHD engineering computations. For incompressible MHD, we propose the fully discrete finite element exterior calculus (FEEC) method that simultaneously and exactly preserves key physical properties-including mass conservation, magnetic flux conservation, current density conservation, energy conservation, and the conservation of magnetic helicity and fluid helicity-even under their respective physical limits. We also develop a linear solution scheme that supports the conservation of MHD helicity and construct an efficient and robust MHD preconditioning solver tailored for the physically preserving discrete system. Numerical experiments validate the accuracy, stability, and robustness of the proposed method under extreme physical parameters while maintaining all these physical properties. Test cases include the Orszag-Tang vortex and several benchmark problems for driven magnetic reconnection, with fluid and magnetic Reynolds numbers simultaneously reaching as high as 10^6. For the ideal compressible MHD system, we combine differential forms and Lie derivatives to define a generalized material derivative, establishing a new Lagrangian formulation of the ideal MHD equations based on this concept. Building upon this formulation, we develop a Lagrangian FEEC algorithm that achieves high-order accuracy in both space and time, preserves positivity of density, ensures a divergence-free magnetic field, and conserves magnetic helicity. To address mesh distortion issues, we design and develop a structure-preserving, helicity-conserving arbitrary Lagrangian–Eulerian (ALE) FEEC algorithm.