How quickly does ethanol evaporate?
Certainly water and ethanol and all mixtures thereof will completely evaporate without boiling given enough time.  

Water and ethanol form a postive azeotrope that is 95.6% ethanol and 4.4% water and boils at 78.2 degrees C.

If you boil a mixture of water-ethanol that has less than 95.6% ethanol, the concentration of water in the liquid phase will increase, but it will only asymptotically approach pure water as all the liquid boils away.  There will not be a time when all the ethanol is gone and you are left with a finite amount of pure water.  

If you boil a mixture of water-ethanol that has more than 95.6% ethanol, the concentration of ethanol is the liquid phase will increase.