Neutral electrons do exist, they are called electron neutrinos. They are part of what is called the "Standard Model" particle physics.

The Standard Model says that are a limited set of fundamental particles that make up all the matter and interactions in the universe. Electrons and electron neutrinos are part of the standard model, as well as quarks, which make up the nucleus of atoms, and photons.

Photons are actually neutral charge, but they do mediate the interactions between positive and negative charges.

The Sun is made up of mostly hydrogen and helium and is currently undergoing fusion. When all the hydrogen is consumed, it will turn into a white dwarf, which is mostly carbon and oxygen. Stars that are bigger than our Sun can create a supernova when their fusion stops, which leaves behind a neutron star. A neutron star is much smaller than earth, but can have about as much mass as our Sun. This means it is super dense! Neutron stars are very hot on the surface (600,000 K), but are not undergoing fusion, so they are not producing energy.