Given a 3D unstructured grid consisting of mixed types of shapes (hex, tet, ...), is there a method to know how many faces (including boundary faces) are contained in the grid?
You might be interested in Euler's characteristic, which relates the number of vertices, edges and faces in a planar graph, that you can also interpret as a 2D mesh. There is a similar relation for 3D meshes (see for example here (section about tetrahedral meshes) or here. The problem is that in 3 dimensions, this (single) formula relates the number of vertices, edges, faces and elements in your mesh. Most mesh generators give you immediately number of elements and vertices, but not the number of faces and edges, so you are left with one equation and two unknowns. I'm afraid there's no easier way than counting the edges or faces.