First, it is definitely theoretically possible to have a seven-sided die. The die as shown is a pentagonal prism. Geometrically, the die’s fairness is most strongly affected by the size ratio of the pentagonal faces to the rectangular sides.
Do nine sided dice exist?
Nine-sided d9 die. This design is based on spacing points as equally as possible on a sphere and then cutting planar slices perpendicular to those directions.