Distortion is not a factor once the panorama is stitched. When making a 360x180 panorama it is not important what lens is used it can be a regular Rectilinear, Tilt, Wide Angle, Circular Fisheye or Fullframe Fisheye it makes no difference at all.
Sure if you shoot with a fisheye you will see curved lines (before stitching) that would normally display straight but once stitched all distortion is corrected and you cannot tell the difference at all.
If you see distortion in a panorama VR (virtual reality) it is because of a poorly set FOV (field of view) setting meaning the view shown is far too wide and outside the scope of the normal human eye view and perpective. These FOV settings are fully adjustable within the software.
It should also be noted that any fisheye (or wide angle) image for that matter can be "de-fished" and perspective corrected.
Shown in this example is a Canon 15mm Fisheye images before (as shot with fisheye) and after ( having been de-fished):
So to answer your question:
So does the Sigma 8mm, 10mm FE, Tokin or Nikkors listed have fisheye distortion?
The answer is yes! But as I have explained it is of no consequence.
The advantage of the fisheye lens is that it allows you to shoot a full 360x180 panorama with minimal shots.
Now there are variables, it comes down to the camera body being used and the size of it's sensor and then the (fisheye) lens being used and in the case of the Tokina 10-17mm at what zoom level.
Here is some common set ups:
Canon APS-C small sensor DSLR cameras (I.E: 500D, 50D etc) or Nikon DX small sensor DSLR cameras (I.E: D90, D300 etc).
When fitted with Sigma 8mm Fisheye: Shoot 4 around at 90 degree intervals with +4-7 degree tilt up to close the zenith (top) and than shoot handheld nadir (down). Or you can shoot 4 around at 90 degree intervals and 0 degree tilt with 1x Zenith at +90 degrees and 1x Nadir -90 (or hand held -90).
When used with a Sigma 10mm Fisheye or Nikkor 10.5mm or Tokina 10-17mm Zoom Fisheye at 10mm: Shoot 6 around at 60 degree intervals with -15 degrees tilt down (this will minimise the tripod) 1x +65 degrees zenith and 1x -90 nadir shot for patching. Or you can shoot 6 around at 0 degrees with 1x Zenith at +90 degrees and 1x Nadir -90 (or hand held -90).
Full size sensor cameras such as Canon 5D, 5DMKII, 1Ds etc or Nikon (FX Sensor) D700, D3, D3x or Sony's full size A900 DSLR:
When fitted with Sigma 8mm Fisheye: Shoot 3 around at 120 degree intervals with +4-7 degree tilt up to close the zenith (top) and than shoot handheld nadir (down). Or you can shoot 3 around at 120 degrees and 0 degrees tilt with 1x Zenith at +90 degrees and 1x Nadir -90 (or hand held -90).
When used with a Sigma 10mm Fisheye or Nikkor 10.5mm (shaved) or Tokina 10-17mm Zoom Fisheye (shaved) at 10mm: Shoot 3 around at 120 degree intervals with +4-7 degrees tilt up and 1x -90 nadir (hand