Answer

There is no connection at all between the questions. The prohibition on the grounds of building refers to activating the device or the electrical circuit in the home. By contrast, a prohibition regarding electricity that is produced on the Sabbath refers to a situation in which the activation itself does not involve a prohibition. The problem is the production, not the use in itself. There is certainly reason to use a generator (that is what I did when I was in a place where that was possible), but strictly speaking, in my opinion, it should not be forbidden. Nowadays it is not possible to shut off the electricity on the Sabbath, and from that point on at most there is only an increase in quantity beyond the minimum needed, and even that is not really the case, since there is no human involvement in production except in cases of malfunctions.