Does grounding to earth make a circuit (allowing electricity to flow)?
Why then would you get electrocuted if you touched a live wire while
  grounded, since it is not a proper circuit?


Because it IS a proper circuit. 

If the return wire of the high voltage source is grounded then, touching the live wire will close a circuit where your body (also grounded as per the quote above) is the "load" and a current will flow.

If the high voltage source were floating i.e. it didn't make a connection to ground then you might feel a little tingle when first touching the live wire but this is due to discharging the small amount of capacitance that the floating voltage source has (with respect to ground).


  Also, when the single terminal of a flyback transformer (usually the
  anode cap on a CRT monitor) is placed near the ground, it creates a
  bright arc to the ground. How does that happen if it's not a complete
  circuit?


Usually, the high voltage winding of a CRT flyback transformer is grounded (or partially grounded via a resistor) and therefore the arc is completing a circuit.