While SFO isn't in the city limits, it is owned by the City and is in an unincorporated section of San Mateo; the SF police have a significant presence (probably the largest force) on site.
Yes they do. And some also call it Frisco and San Fran.
After 35 years in the Bay Area I have settled on calling it "The City" but will bust out with "San Fran Frisco" on special occasions to just to be irritating to those who desperately want to "fit in."
And the SF office of Google is SFO. But nobody calls the city SFO.