Is the dual of a polyhedral embedding a polyhedral embedding?
You need to specify that $G$ is a simple graph (else the trigonal hosohedron is a counterexample) but then the answer is yes. What can go wrong? If a dual face is not a simple cycle then one of its edges is repeated, giving a self-loop in the primal, or a vertex but not an edge is repeated, in which case this face and vertex form a configuration of the same type in the primal. If two dual faces meet at more than one edge then there is a multiple adjacency in the primal. If the dual has a self-loop or multiple adjacency then we get a face with a repeated edge or two dual faces meeting at more than one edge in the primal. If two dual faces meet at two vertices but do not share an edge between these two vertices then a configuration of the same type happens in the primal. And if two dual faces meet in an edge and a third vertex then in the primal we have three faces sharing the same two vertices and they can't all three contain the edge connecting these two vertices.