The Gilded Cage was a movie theater located at 628 Broadway, just down the street from the Broadway (815 Broadway). It was opened around 1966/7 and didn't have a very long lifespan. As far as I can tell, it was built out of a storeroom.
Unlike many of the downtown theaters, the Plaza, Cabrillo and Broadway, the building that contained the Gilded Cage is still standing. It has been remodeled several times since so any trace of it has long since vanished.
As for the kinds of movies that played there?
Well, the morally instructive ones that dealt with gripping social issues of the day. And they had boobies.

I'm also including the daily ads for the Center and Cinerama.

San Diego Tribune. Wednesday 13 December 1967.