Reference signal when measuring EEG
For the unipolar configuration, why can't we just use Earth ground as
  our reference instead of having a reference electrode or performing an
  average of measurements on all the electrodes?


The ground has a different function than the reference. The reference is the electrode that the voltage is recorded against; a voltage is a potential difference so you need two electrodes. The ground is used for common mode rejection. The primary purpose of the ground is to prevent power line noise from interfering with the small biopotential signals of interest (source: Biopac). 


  For the bipolar configuration, if the electrodes are too close to each
  other, wouldn't we measure very low voltages since much of the signal
  appearing on one electrode would also be picked up by the other?


Yes. The smaller the distance, the better the rejection of unwanted signals. But with that, you're also risk to throw away the baby with the bath water.