Why did my lemon and penny battery fail?
By not removing every last trace of copper from the sanded-side of the pennies, I created a kind of short-circuit that discouraged the charges from moving through the stack of pennies.
  


This is the main problem. Until all of the copper is gone from the zinc side of each penny, there is little reason for the two halves of the electrochemical reaction between zinc and copper not to just take place locally.  Your battery likely had one or more ionic short-circuits, just as you say, on these incompletely sanded surfaces.


  
  Since the cardboard was not very absorbent, it might have been particularly porous either and this might have prevented sufficiently free flow of particles between each penny in the stack.
  


This is unlikely.  As long as the cardboard was soaked through with the lemon juice, it should function fine as an ion-transport medium.


  
  By using a relatively low number of pennies, I guess I set it up to have a relatively low voltage. Maybe I could have overcome problems 1 or 2 if I used more pennies.
  


This is also unlikely. With five junctions (one between each neighboring pair of pennies), at a theoretical $1.1~\mathrm V$ per junction your stack voltage should have been at least about $5~\mathrm V$. This should be plenty to light a simple LED.