When Homer runs its simulation does it take in consideration the wind turbine height?

Yes, HOMER takes the hub height (tower height) into account when calculating the power output of the wind turbine.  You enter the anemometer height and the wind shear properties in the Wind Resource Inputs window.  You enter the turbine hub height in the Wind Turbine Inputs window.  Every time step, HOMER uses the wind shear properties to estimate the wind speed at hub height.  Please look up "wind turbine power output" in the index of the Help file for details.