Using a significantly large metallic sphere, could we draw out the electrons from the negative terminal of a battery? I've pored over many articles on electrochemical cells, and I understand that on the oxidation side of a zinc-copper Daniell cell, the anode is highly negatively charged because zinc participates in oxidation and loses electrons as the ions go into the zinc sulfate.

So since the anode is rich in free electrons, is it possible to take an electrically neutral, large conductor (a metallic sphere for example), touch it to the negative terminal of the battery so that the free electrons will distribute themselves over its surface, discharge the sphere, and repeat this until the negative terminal is neutralized?
Yes, this is possible, you have effectively described using a battery to charge a very weak capacitor, then discharging the capacitor.  The charge stored will be minimal (possibly immeasurable), pico-coulombs or less and the number of cycles to fully discharge then battery may be on the order of billions or trillions, but in theory the process described is possible (more properly: plausible).