Do you mean NSA (National Security Agency) or NASA (National Aeronautics and Space Administration). Eh, doesn't matter. The United States doesn't have a "Copy Right" on any product of any agency or agencies themselves. They might not like certain stories about their agency and may not offer to help out with the big budget blockbuster if the film is critical of the military or your name is Michael Bay (Most of the military equipment used in the Transformers live action films and the soldiers featured in the background are U.S. Military equipment and personnel. Movies and shows that portray the military in a good light can apply for Pentagon Backing where they will charge the production company for the cost of the use of the vehicles but will do it in the most authentic way possible. The reason most of the villains in Bay's films are also U.S. Military vehicles is because Bay basically convinced them that the villains wanted the most bad-ass Earth vehicles and the U.S. Military was the obvious source for that requirement.).
Generally all Three Letter U.S. Government organizations will be available to critique as will most Western Liberal Democracies governments. The more testy governments will just ban the books from sale in the nation.
The Vatican is a legitimate country, so they cannot be off limits either and like you pointed out the Vatican Archives are the stuff of fictional legend (The real archives are more mundane than the Hollywood version... usually they are off limits for preservation of documents from the earlier part of the church's 2,000+ year history).
Secret Police tend to vary between fake and real, because fictional nations give some countries Secret Police when that real world country doesn't have such an organization.
Generally, fiction depends a lot on who is the hero... If the Leader is going to be involved, it will usually be a fictional leader unless unavoidable (The Queen of England is usually Elizabeth II, but the President of the United States is usually always fictional. The Pope is always the Pope and always Italian (even though there hasn't been an Italian Pope since the Mid-70s in real life... the last three were a Pole, a German, and an Argentinian. Though prior to those three, the Pope was Italian.). Fictional U.S. Presidents tend to be pretty moderate, though they can also be reflective of the current sitting president or a favorite historical one.
If their appearance amounts to stock footage (ala Forrest Gump) then use the historically appropriate one.
Anyone directly involved with the agencies will usually be fictional representations (unless it's J. Edgar Hoover, and he was in charge of the FBI at the time). Though mimicking real historically famous employees is always an option.