When Don and Rachel are on her couch, and he says, "No, not unless you tell me you want this", her head is on the sofa armrest. Suddenly her head is instead on the red pillow and she says, "yes please".
When Bertram Cooper asks Joan to help send out telegrams, he dictates names and addresses while she sits at a typewriter typing telegrams to be sent. In real life, telegrams were created either one of two ways: going to the Western Union office and writing the telegram or telephoning Western Union and dictating it. They were not typed out and mailed like postal mail, the way this scene intimates. To be period accurate Joan should have picked up the telephone and read off names of recipients and the body of the message.
While watching a film of campaign ads, Don tells Harry to turn off the projector in the middle of a Nixon ad. Minutes later, when Harry is removing the film from the projector, the film is completely transparent with no picture.