Is the gulf stream both a wind-driven current and part of thermohaline circulation?
Yes.Thermohaline circulation is dependent on differences in water density, caused by differences in temperature and salinity between different bodies of water. It therefore requires deepwater areas, of which the northern and middle Atlantic have the most, for these differences or "gradients" to form. These are influences on the Gulf Stream.

In addition, the Gulf Stream is wind driven, as is true of most currents. So it depends on both influences.