Statistics - Does the majority of new pilot training occur at controlled or uncontrolled airports?
No one really tracks these stats and they would be tough to track on any note. The latter half of my training was out of an uncontrolled field but lots of touch and gos occurred at controlled fields so I had lots of experience talking to the tower even though I was not flying out of a towered field.  


  I'm interested in this because it would likely give an indication of
  the relative level of comfort of new pilots operating in both
  controlled and uncontrolled airport environments.


As a pilot you are responsible (as per the ACS) for understanding the regulations and procedures for both situations and in todays day and age it would be foolish for an instructor to not at least present both situations in your training. That being said, you can technically get your PPL in an airplane that does not even have a radio but you are still responsible for understanding the regulations and may be asked to explain them on your check ride. 


  I assume that it's likely that a Designated Pilot Examiner would be
  more likely to operate out of a controlled field just due to traffic


The DPE that I went to was at an uncontrolled field (as are the other 2 in my area). Im not sure this is an accurate generalization. 

Keep in mind the DPE will go over your logbook and may look at the airports you landed at. If you dont have sufficient towered experience they may ask extra questions about that but you may not be required to demonstrate it on your check ride.