Why can't an excess of electrons or holes by themselves cause current flow? I am a beginner in electrical engineering. Often times (most cases actually), the underlying physics aren't really explained to us and we are just left to assume that it works "because it works." This is never enough for me in classes etc and I always end up doing a follow up of physics side of the spectrum.
My question is, you have a battery with excess electrons built up on the negative lead, and excess holes on the positive lead, why is it that in our universe then connecting a conductive compound (like.. copper) to just the negative lead does not produce current, or vice versa connecting the wire to just the positive lead. 
After all, if we move a magnet passed a conductor, there is a tiny induced current. How can the magnet do this, but an excess of electrons or electron holes repelling each other cannot?
 A: Physicist Plug
You may want to consider becoming a Physicist, especially if the ambiguity of engineering classes bother you. :)
Electrons and Holes Need Holes or Electrons to Flow to!
Excess electrons or holes do not cause current to flow because those excess electrons (or holes) need someplace to go to. If the electrons/holes don't "see" any places to go to, you don't get current. You need a path for the electrons and holes to flow. (This path also has to be energetically allowable, that is: the electrons need to have enough energy to follow that path. That's why batteries don't randomly spark and arc.)
Batteries Use Entropy to Make a Potential Difference
It just so happens that many batteries use entropy to keep the holes at one end and the electrons at the other. This trick is complicated, and something you should ask the Chemistry Stack Exchange, or at least in a different question here. Entropy keeps the holes and electrons apart, despite the fact that they would electrically come together.
Additionally, batteries also work by allowing ions (not electrons or holes) to flow from one end of the battery to the other. This allows the electrons to flow through the rest of the circuit instead of having them discharge like capacitors. Once again, this is where chemistry and physics meet, and you should try the chemistry stack exchange for more information.
The Conclusion
Since electrons/holes need holes/electrons to flow to, you're safe touching one end of a battery, but in trouble if you hold both ends. If you're holding only one end of the battery, the holes/electrons don't "see" anywhere to go, so they don't go. If you connect the circuit, the holes and electrons "see" each other at opposite ends of the circuits and then flow through it to meet up. You could also get current if you brought something with a lot of holes or electrons near the appropriate end, although it would likely be in the form of a sudden static discharge instead of the smooth flow from a battery.
Maxwell's Equations, Magnets, and Induced Currents
Finally, if you take a look at Maxwell's equations, you'll find an equation with a term that relies on a change in magnetic fields to produce current. More specifically, it's those last two which explain the relations between magnetic fields and electric fields. Since you're starting your career as an electrical engineer, you may want to wait until you have a good grasp of vector calculus before tackling those.
A: If you connect a piece of wire to just one terminal of a battery, the whole wire tends to gain the same potential as that of the terminal it is connected to, this requires an instantaneous current and is achieved very quickly.
The need of connecting the wire to both terminal, is that the internal chemical mechanism of the battery maintains a potential difference across the wire, thus the wire never reaches a desired potential and continues to try to reach it by flowing current.
The moving magnet also provides a stable potential difference between 2 points on a conductor which just by transference of electrons isn't diminished or nullified, because it is controlled by the moving action of the magnet.
Whereas, the excess/lack of electrons, provide a potential differece only till redistribution takes place and whole conductor gets to the same potential.
A: An excess of electrons can cause a current flow.  
Imagine two spheres of metal.  One is neutrally charged, but the other is negatively charged with an excess of electrons.  Connect a wire between them, and what happens?  The charge re-distributes itself so that the charge on both spheres is equal.  As this is happening, a brief current of electrons is flowing through the wire, but it stops as soon as the charge is equal.

My question is, you have a battery with excess electrons built up on the negative lead, and excess holes on the positive lead, why is it that in our universe then connecting a conductive compound (like.. copper) to just the negative lead does not produce current

It does produce a current.  It's just very small, because the voltage of the battery and capacitance of the wire are very small, and it's very brief, because there's no path for the current to flow continuously, so after the charge has equalized there's no more current.
