Do the elevators generate vortices?
Yes, they do. The elevator vortices I have observed are pinned to the trailing edge corners of the elevator and are made visible when the aircraft is traveling through moist air that is very close to the dew point temperature, and at a significant angle of attack. There's a hotel at LAX where you can look out the windows and watch planes fly by on final, just before touchdown; on a foggy afternoon they are all trailing vortices behind them.