While Malcolm Tucker might consider "climb the mountain of conflict" would be something a "Nazi Julie Andrews" might say, it was Marjorie McKay (ghosting for Peggy Wood, who played the Mother Abbess) and not Julie Andrews who sang "Climb Ev'ry Mountain" in The Sound of Music (1965).
In the scene where Malcolm is walking outside the White House talking on the phone, he walks a segment from a light pole to a police car with very few people around. However, when Judy hangs up and the camera switches to face the fat man, Malcolm is back in the first light pole and the place is suddenly crowded. Then the camera turns around and shows the police car ahead which is now surrounded by a group of people on Segways.
When Malcolm Tucker leaves the White House urgently for the State Department, he is seen running eastwards on Pennsylvania Avenue, and in the following scene past the Willard Hotel on E St NW. This route would take him directly away from the State Department, which is west of the White House.
Simon and Toby land in DC at what seems to be the afternoon (it is still full daylight, there are no long shadows and they proceed to the hotel and discuss what they'll do that evening). They phone Judy in London to arrange transport for them. She is woken by the call in what she says is the middle of the night. But DC is only 5 hours behind London, so it should be no later than mid-evening there.
General Miller (James Gandolfini) is wearing a unit patch on his left shoulder. However, officers assigned to the Pentagon, since they are not officially assigned to units, do not wear a shoulder patch. At the apparent time of the movie (pre-Iraq), even the current generic square US Army patch was not as yet authorized.