So all observed reactions are a result of heat
In most cases - yes.
If electricity is the result of the charge of subatomic particles, how exactly is a release of it in an exothermic reaction a result of heat?
In many (but not all) reactions electrons have to move from one molecule (atom, ion) to another. If we can separate the reactants and force electrons to flow not directly between them, but through some external circuit, we have a battery.
And light, I've learned, is when the electrons move closer to the nucleus, and thus the energy released in that instance is light. How is this light released a result of heat? And how exactly does moving closer to the nucleus produce light?
Electrons on "lower" orbits have lower energy than those higher up. When electron goes down (jumps from the higher energy orbital to the lower energy orbital) it has to get rid of the excess energy and it emits it as a photon - electromagnetic radiation. Depending on the amount of energy involved wavelength of the photon will be different - it can be visible light of any color, it can be UV or even X-Ray.
This is not exactly energy given as heat, we can say energy was given away as light (or radiation).
As you are willing to get deeper... Exothermic reactions give energy. This energy can be in form of heat (that is, kinetic movements of molecules - the hotter the substance, the faster the molecules are), or radiation - like light. Sometimes one of the form of energy is more prevalent, sometimes the other. However, hot substances can emit radiation as well - this is so called black body radiation. While the mechanisms of emission are quite different, effect is in a way similar - you have something that radiates. If something gets hot enough it will emit visible light - that's what happens in the fire. At the same time radiation can get absorbed and the substance can get hot (that's what you see whenever Sun shines and heats up everything). So, you see, it is all connected, and while there are several mechanisms involved, they are all based on the same molecules, moving electrons and orbitals - there are no "separate particles" responsible for light, sound, fire and so on -which was starting point of the discussion.
Fire seems simple, but what causes the actual flame, the flame we see? Could it possibly be a combination of a temperature and a light release?
See above. Black body radiation is a key term in flame. If you light a candle light is generated by microscopic particles of soot heated up in flame.
It is possible to emit light without a high temperature, if we can excite atoms (molecules) so that electrons go up to higher energy orbital and the fall back, emitting light. But that's not what is happening in flame (although to some extent it is - yellow color observed in flames is often effect of the sodium atoms being excited and deexcited).
Disclaimer: I currently am running off of Bohr's Model of the Atom. Please enlighten me on any new features that I am blissfully unaware of, if it is an answer to my questions.
Also: I understand that my mind MAY explode with the revelations I receive about the atom. I have a splash mat ready...
You are trying to run before learning how to walk. It won't work. But you seem to be curious, that should get you there