Cosmology noob here.
Since we have observable and documented evidence that galaxies, including our own, are rapidly moving away from each other, has there been any attempt by scientists to plot the exact location in 3-dimensional space where the "big bang" actually occurred?
Everywhere. It's not an explosion, its a rapidly expanding point.
Your question is like blowing up a balloon and asking "what us the exact location of the balloon when it was less inflated?
No, the cosmological constant is weaker than all other forces. Protons and neutrons are a small bit larger than you would expect from just measuring strong and electromagnetic forces, but the expansion is overhwlemed by gravity, etc.
In between galaxies, where there is no real concentration of matter or energy, the cosmological constant dominates, and thus space expands
...And The Lord God Almighty owns the balloon factory.
Don't forget that part.
And yet, at every point, there is a center of the balloon's sphere. The point which is farthest from all the edges.
Thanks for that visual it was quite helpful I totally didn't know what you meant by center of the balloons sphere.
You're asking a question that is equivalent to "which point on the edge of a sphere is farthest from every edge?" There is nowhere on the universe that is the center of the expansion. It is outside of our observations, or "inside" the universe by your analogy.