"...How does overheating or certain ingredients cause the lye-water and oils to separate from one another ?..."
An emulsion is a mixture of tiny droplets of one liquid suspended in another.
Mechanical mixing (stick blender or hand stirring) is required to create an emulsion. The smaller the droplet size from the mixing, the more likely the emulsion will stay together. An example of a mechanically mixed emulsion is a vinaigrette made with only vinegar and oil. Even if you mix it in a blender, the mixture will quickly separate out after you stop mixing.
Mixing is only part of the story.
For long-term stability of an emulsion, you have to also use a chemical emulsifier. In soapmaking, the soap molecules are the emulsifier that stabilizes the batter emulsion. The more soap molecules in the batter, the more stable the emulsion will be. An example of this type of emulsion is a vinaigrette made with oil, vinegar, and mustard. The mustard is the chemical emulsifier that keeps the salad dressing well mixed.
In soap making, "trace" is a visual clue that tells you there are enough soap molecules in the batter to chemically stabilize the emulsion, and it's reasonably safe to stop mixing.
Part of the art of soap making is to recognize when this safe point is reached and one move on to the next steps in the soap making process. If you mix too long before pouring your soap into the mold, the soap can get very thick due to the increasing numbers of soap molecules. The batter may even solidify in the pot.
On the other hand, if you stop mixing too soon, you may get separation in the mold because you don't have enough soap molecules in the batter to stabilize the emulsion. Soapers who want to make fancy swirls will stop mixing at the point where the batter is just mechanically emulsified. They are counting on the growing number of soap molecules to stabilize their soap batter as they work. It's a bit of a risk, but a necessary one for doing this type of fancy work.
Anything that causes the liquid droplets to coalesce into larger droplets will cause an emulsion to fail, meaning the ingredients separate back into layers.
Excessive heat is one culprit. It can reduce the viscosity (syrupiness) of an emulsion, increase the chances of the liquid droplets coming together into larger droplets, and trigger chemical reactions that cause emulsion failure. If you have ever made a custard of milk and eggs and had it curdle, that is an emulsion failure due to excessive heat.
Another culprit is ingredients that interfere with the chemical emulsifier, with or without added heat. I can't think of an example in soap making at the moment -- maybe someone else can chime in with a good one.