The sharpening setting in the picture control is a software setting that affects the output file, regardless of the lens used.
The sharpness of the lens, when testing, can be ascribed to so many factors, unless you really control for variables. But, given your quick test the results aren't surprising. Generally, super zooms lenses like the SIgma 18-200 are not as sharp as something with less range. Why the 24-120 wasn't as sharp is beyond me, but as it is an unfamiliar lens to you, that may have had something to do with it.
If you are going to do a quick test, I suggest that you use a tripod, are remote release or delayed shutter release and shoot at f/8 at a variety of focal lengths on each lens. That should not take too long and you may find that a good lens at 50mm is not as good as the others elsewhere.
For me, if I were going on a trip with just one lens, I would take a superzoom to have maximum flexibility. My preference for the D7000 would be the new Nikon 18-300 or the Tamron 18-270 VD PZD.