Do electrons go to even lower ground states after exothermic reactions?
Yes, electron energy levels are like that, especially for single atoms.

Yes, forming molecular bonds creates different electron energy levels, with exothermic reactions leading to chemical bonds with lower ground levels then the reagents had.

Note that by an electron ground energy level is meant the minimal energy of  energy levels unocuppied by other electrons.

As an electron is forbidden by quantum rules to be in the same quantum state as any other electron within an atom or molecule.

Thermal, photon or collision excitations may keep part of electrons in excited state.

Much more info you can get reading about 


molecular orbitals 
molecular orbitals diagrams 
molecular orbital theory


on Wikipedia.