The enlisted men in the radar room at the beginning are wearing Army insignia of rank, not Air Force.
When the MiG is shown at the beginning of the film it's not an actual MiG, but a US-built Republic F-84F. In 1956 neither the U.S. Air Force nor the movie studio would have had access to flyable MiGs. The Republic F-84F Thunderstreak was a very close stand-in for the MiG-15 & MiG-17 as it bore a close resemblance to them and so was used in various movies of that period. The U.S. aircraft sent out to intercept it are North American F-86Ds.
When questioned by Katherine Hepburn as to how fast he had flown, Bo Hope states 500 MPH. The F-86D Sabre that he was flying on the intercept had a maximum speed of 715 mph (1,151 km/h, 621 kn) or Mach .93. A fighter pilot would have put his machine through its paces to see what it was capable of before trusting his life to it. Flying 200 MPH slower than maximum would have been normal for cruising but to assume he would never have pushed it to the maximum or near-max is absurd.
The USAF officers keep referring to the Army. The U.S. Air Force separated from the U.S. Army in 1947. By 1956 they would not have been having daily interactions with the senior service.
The MiG shown at the beginning of the film has the letter S written on it. A Soviet aircraft would only use letters from the Cyrillic, not the Latin, alphabet.
When Lockwood and Kovelenko arrive at the "USAF Control Office" the background to the left with the old car parked in front of a building is obviously a flat, painted backdrop and not real or a part of a set. There is no depth or shadow.
When in Colonel Tarbell's office, a boom mic reflected in the window (an Interior Transom) over the office door, behind Vinka (Katharine Hepburn) who is seated.
In 1956 when the movie was made, Bob Hope was about 53 and Katharine Hepburn was around 49. This would have been way too old for two front-line pilots with the ranks of Captain in either Air Force. 1950's fighter pilots would likely have been in their 20's or at the outside in the early 30's.
In the quotes posted Bob Hope's character is listed as "Major". In the movie he is listed as "Captain" and wears the rank of an Air Force Captain.