Enshrouded in several well-known controversies, dwarf galaxies have been extensively studied to learn about the underlying cosmology, notwithstanding that physical processes regulating their properties are poorly understood. To shed light on these processes, we introduce the Pandora suite of 17 high-resolution (3.5 parsec half-cell side) dwarf galaxy formation cosmological simulations. Commencing with thermo-turbulent star formation and mechanical supernova feedback, we gradually increase the complexity of physics incorporated leading to full-physics models combining magnetism, on-the-fly radiative transfer and the corresponding stellar photoheating, and SN-accelerated cosmic rays. We investigate combinations of these processes, comparing them with observations to constrain what are the main mechanisms determining dwarf galaxy properties. We find hydrodynamical `SN feedback-only' simulations struggle to produce realistic dwarf galaxies, leading either to overquenched or too centrally concentrated, dispersion dominated systems when compared to observed field dwarfs. Accounting for radiation with cosmic rays results in extended and rotationally-supported systems. Spatially `distributed' feedback leads to realistic stellar and HI masses as well as kinematics. Furthermore, resolved kinematic maps of our full-physics models predict kinematically distinct clumps and kinematic misalignments of stars, HI and HII after star formation events. Episodic star formation combined with its associated feedback induces more core-like dark matter central profiles, which our `SN feedback-only' models struggle to achieve. Our results demonstrate the complexity of physical processes required to capture realistic dwarf galaxy properties, making tangible predictions for integral field unit surveys, radio synchrotron emission, and for galaxy and multi-phase interstellar medium properties that JWST will probe.