I assume you mean from the classical electrodynamics side only, not from quantum electrodynamics side. By the former, even H atom cannot exist, as the electron would fall along a spiral curve on the nucleus, continually emitting radiation being radially accelerated.
For a hydrogen molecule, both electrons move around both protons, they are not dedicated to ...
Orbitals have negative energy, so closer to zero means higher energy. It is analogous to the energy of orbiting celestial objects.
Orbital energy is the energy of an electron in quantum state related to the orbital.