How are faces of a DCEL updated when triangulating a polygon?
For the algorithm that triangulates a monotone polygon: every time you add a diagonal, you actually find a triangle. So, you can indeed explicitly relabel the face-value associated with all edges of that triangle. That does not affect the asymptotic running time. 

For the algorithm that splits a simple polygon in to monotone polygons you cannot really afford that, as you may have to relabel an (half)edge multiple times. Hence, a solution there is just to be lazy: i.e. use the sweep just to update the vertex/edge fields in the DCEL, and ignore the face values. When you are done with the sweep: use the vertex/edge fields, in particular the next-fields on the half-edges, to walk around every face, and set the face values appropriately.