Lock Martin, the doorman at Grauman's Chinese Theater, was cast because of his nearly seven-foot height; however, he was not a physically strong man and could not actually carry Patricia Neal, so he had to be aided by wires (in shots from the back where he's carrying her (actually a lightweight dummy in his arms). He also had difficulty with the heavy Gort suit and could only stay in it for about a half hour at a time.
The equations seen on Professor Barnhardt's blackboard are authentic physics that describe a particular form of the famous "three-body problem" in Newtonian gravitation, a problem which has no general closed-form solution. Such many-body problems are central to navigation in interstellar space. The manner of writing and organization of the terms show that a real physicist had produced the work.
To give the appearance of seamlessness to the space ship, the crack around the door was filled with putty, then painted over. When the door opened the putty was torn apart, making the door seem to simply appear.
Patricia Neal admitted in interviews that she was completely unaware during the filming that the film would turn out so well and become one of the great science-fiction classics of all time. She assumed it would be just another one of the then-current and rather trashy flying saucer films and found it difficult to keep a straight face while saying her lines.
To increase the sense of reality, some of the most famous broadcast journalists of the time were hired to do cameos as themselves. These included Gabriel Heatter, H.V. Kaltenborn, Drew Pearson, and Elmer Davis.