The main foreign government building is actually Pasadena City Hall, used for many movie locations and by famous productions including Charlie Chaplin's. "The Great Dictator." It is perhaps most famously seen as the City Hall in the TV show Parks And Recreation.
No mission instructions are given to Jim Phelps in the opening scene.
Writers sometimes use character names for subtle ironic purposes. Rollin, in disguise, tells Kurtz that Jim Phelps is Colonel Geist. Geist is German for "ghost" or "phantom". Such words, including "spook" are euphemisms for spies.
In the sequence where Will Kuluva (Rudolf Kurtz) is hidden in the box and made to believe that he is being transported to the "East Zone" the tape recorder used to play the sound effects is actually a Nagra-brand tape recorder. This was the standard sound recording system for the film industry and television production from the 1960s until the 1990s.