Can we eliminate adjacent opposing pairs while classifying the surface in other than the second step? When we apply the method of classification of surfaces to a given surface, in the second step we can eliminate adjacent opposing pair, but can we do this when, for example, in the sixth step if we get any adjacent opposing pair like $$.. d d^{-1}... ?$$ Note that, by saying that _...th step_ , I'm referring to the constructive proof of the classification of closed & compact surfaces theorem in Kinsey's _Topology of Surfaces_ book.

Yes, you are taking a connect sum with a sphere, which topologically does not change your surface.