Question: Why was the song "My Blue Heaven" used so much in the first season?
Answer: My guess is because Fats Domino's version was #17 on Billboard's charts for 1957. It helps identify the time period. And, I think, at the time copyright laws only protected material for 25 years after release so the song was free to use. Now, material is protected for 75 years.
The Korean War was fought between 1950-53. The song didn't exist yet.
Question: Why do the helicopters fly backwards? Is that just a filming mistake or did they do it for a reason?
Chosen answer: The helicopters aren't flying backwards, they are simply not moving through the camera view as fast as the camera is moving.
Answer: The copyright information provided in the answer is wrong. Although there's a lot more to it, works published before 1978 are protected for 95 years (28 years for the initial term and 67 for the renewal term). "My Blue Heaven"'s copyrights were renewed in 1953 and 1957. Works published after 1977 are protected for the life of the artist plus 70 years. It should be noted that the movie "My Blue Heaven", starring Betty Grable, came out in Dec 1950 in which the song was sung during the credits, which would have been during the Korean War (not that I'm saying that's the reason for its use).
Bishop73