Ja problemslos

This works without any problems. With regard to camera shake, a tripod or other support is recommended in any case (even on a camera with a full-frame sensor) to obtain sharp images. Since the lens loses some sharpness towards the edges, especially at 600 mm focal length on the larger full-frame sensor, it is an advantage that the D7100 with the smaller sensor (--> smaller angle of view) only uses the centre of the lens for the shots. Whether the difference is noticeable in everyday life remains to be seen... If you can live with the weight, there is nothing against using the lens on a D7100 (or other camera with an APS-C sensor).

Yes the lens fits the Nikon D7100 camera, but it is designed for a full-frame camera (e.g. D4, D800, D810 etc.).
Since the Nikon D7100 camera is not a full-frame camera but a camera with a smaller sensor, the Sigma 150-600mm (full-frame) becomes a 225-900mm lens on the D7100.
This means that only a small part of the zoom range can be used for blur-free shots without a tripod, also in connection with the considerable weight of the lens.