I can't test this so I have to ask: If I use shift to shoot a row of 3 images (max shift left, center and max shift right) do I get parallax problems?
Yes, when there is foreground detail then parallax will happen, unless you counter shift the camera by the same amount as you shifted the lens. The goal is to keep the entrance pupil (NPP) of the lens stationary.
Am I correct to assume that if I use NPP rotation and tilt to correct the perspective I'd not get parallax problems? How about any other problems?
Assuming that also the tilt keeps the NPP stationary, then the stitch will remain without parallax. BTW tilt in a Tilt Shift lens is for scheimpflug focal plane adjustment, shift is for perspective. Perhaps you mean tilting the camera + attached lens. Then you probably will move the NPP, and get parallax.
Then thir scenario: Will there be parallax or other problems if I use a normal non-TS lens, NPP rotation and point the lens (around the NPP) slightly down to get more foreground and then shoot a row?
As long as the rotation is around the axes going through the NPP, there will be no problem. You may need a stitching application where you can set the horizon manually, depending on how the stitcher works.