I’m not sure if it can be said that Farley Granger was “openly” gay at the time. Arthur Laurents had a long-term relationship with Granger and wrote about it in some detail in his autobiography. IIRC, they had been a couple for several years when Rope was made, but Granger was still occasionally going out with girls for photo-op “dates” set up by the studio… although I could be wrong about this, since it’s been a few years since I read Laurents’ book. Having said that, those in their social circles and even in Hollywood circles generally knew about Granger, but that was strictly in the industry, and not known by the general public per se – again IIRC.
There are actually two intentionally visible cuts in the main part of the movie: once when the maid enters to announce a telephone call, and again after James Stewart finishes his monologue on how the murders were committed.
No, he’s openly gay now. No one was openly gay back then, except maybe Billy Haines! But he was out in that his friends and coworkers (and Hitchcock) knew he was gay. They certainly weren’t going to announce it to the moviegoing public!