What is the nature of chemical energy?
Chemical energy is really nothing but electric potential energy.

The crucial difference is that chemical energy involves energy that can be extracted by breaking/making bonds.

For example, we say that the reaction $\ce{2H2 + O2-&gt;H2O}$ is accompanied by a release of energy. This energy is the difference of potential energies of the bonds in the reactants and products. Which potential energies? The bond potential energies. Bonds are, at the most basic level, formed due to attraction and repulsion of electrons and nuclei (coupled with some quantum mechanical effects). This carries some potential energy. The change in this potential energy achieved by breaking the bond is the "bond energy", and forms part of your "chemical energy". Some other parts are lattice energy, hydration energy etc, which are (in the end), due to electromagnetic forces/electromagnetic potential.

So yeah, chemical energy is just a "subset" of potential energy,


  If chemical energy is an "own" form of energy, why does it not have a specific equation?


Since chemical energy is an amalgamation of various types of electric potential energy in atoms, it cannot be given a formula (though you can calculate it for some compounds if you know some standard values). 


  Is chemical energy an energy that is "conserved" and if so, why is chemical energy not just potential energy?


Chemical energy isn't really conserved anywhere unless there is no change to the chemical composition of the system. However, in a chemical reaction, most of the time any changes in energy are compensated for by absorption/release of heat. But it is just potential energy, as mentioned.


  Did we have chemical energy before the big bang or when did it start?


Meaningless question, there was no "before" the big bang. Also, energy really isn't something that you should classify into compartmentalized types.

But, since atoms formed a few minutes after the Big Bang, there wasn't anything to facilitate the existence of any chemical energy -- so the first occurrence of chemical energy was then.


  Can you have chemical energy at or close enough to absolute zero?


Yep. Bonds exist at absolute zero. Note that we can never reach absolute zero, but bonds exist both close to and hypothetically "at" absolute zero.


  How much of the total energy in the universe is chemical energy?


I can't answer that, and I doubt it can be answered to any reasonable approximation. I think the percentage would be rather low -- our universe is mainly stars (besides being empty space). The average star doesn't have many chemical bonds--it contains hydrogen/helium plasma, and plasma can't exactly form bonds.


  What's the point of many different definitions of energy?


In certain situations, you know that there is no method to facilitate conversion of one form of energy into another. In such a case, one can separately conserve that form of energy. So the forms of energy help in analyzing/solving a situation, but not much more.