Is this a hypergraph?
No, this isn't a hypergraph.  I don't think it has a standard "name".  Call it whatever you like: as long as you define your terms clearly, you should be fine.

Yes, you can implement this efficiently using any standard data structure for representing graphs: e.g., adjacency lists.  Each node of the "outer" graph will contain a pointer to a data structure that describes the "inner" graph it represents.