No, because the group G of rotations of the 2-sphere that preserve the set of all "dimples" X (so for all g in G, g(X) = X) on the sphere has to have a member g(x,y) in G for all x, y in x such that the action of g(x,y)x = y; that is, any point can be transformed into any other point by a rotation that nonetheless preserves the whole lattice. This is called face-transitivity* and there are a finite number of recursively enumerable sets of face-transitive solids: the bipyramids, the trapezohedra, the prisms and antiprisms (as log dice), and the Catalan solids. Bipyramids and trapezohedra are quite impractical at high face counts because they take on a biconical shape whereas spheroids are ideal. Log dice are fair if they roll a lot, but they're not as cool.

Yes, but the currently-manufactured ones tend to be unfair: https://en.wikipedia.org/wiki/Zocchihedron

The advantage of polyhedral dice is that congruency of faces implies fairness.