Jelly and then the Doctor say 'consiglieri' meaning the
doctor's would-be position in Vitti's family. However, the correct form would be 'consigliere' that is an advisor in Italian. The word 'consiglieri' is the plural from 'consigliere'.
When Vitti is getting ready for the big meeting he sees in his mirror a television ad that reminds him of his father. Before he sees it the TV faces the audience, then must turn to be seen in the mirror. After Vitti sees it, he sits on his bed to watch the commercial, so the TV must again turn back around toward the audience.
FBI agents are shown participating in the 1957 Appalachian raid. In that year, J. Edgar Hoover's FBI's policy was that the Mafia did not exist, so no federal agents were involved in the raid, which was conducted by New York state troopers only.
Prior to the guy falling from the hotel window, there is a lot of gunplay with the guy chasing Jelly and Jimmy down a short hallway in Vitti's suite. As they pass the hall closet, there is a wire visible that pulls the closet open.