Why move the queen’s rook to defend d-file instead of king’s rook? In this position, why is it better to move the queen’s rook to defend the d-file instead of the king’s rook? Moving the king’s rook to the d-file becomes more challenging. Is it important to leave king’s rook behind to support the king? Is control of the f-file more important? I am curious about the logic behind of choosing the queen's rook in such situations.

[FEN "3rr1k1/pp3pbp/1n4p1/4P3/3P2b1/B4N2/P3BPPP/R4RK1 w - - 0 1"]

1. Rad1
White needs to find a way to make use of his extra pawn, because it is hard to see any other plan. His dream would be to play d4-d5-d6, but this is not yet possible because a) d5 is under Blacks control, and b) the Pe5 would hang. Having the Rooks on e1 and d1 would address both of these issues, and that would be a motive to choose Rad1, but there is still play in this position and White needs to be accurate.  Whites K is extremely safe, so that is not a concern. There is no reason to worry about Bxf3. The resulting "weak" pawns do not matter because they cannot be attacked. Best in this situation seems be Bb5 which forces either Bd7 or Nd7, both of which reduce the effectiveness of Blacks blockade. A plausible continuation after 1..Nd7 would be 2. Ng5 threatening Bc4, 2..h6, 3.Ne4 intending Nd6. So Black will have to play ..Bd7, and then White can play Bxd7 and reorganize at leisure. He might consider putting the KR on d1, followed by a Knight move and f4. The QR is then free to occupy the c-file. It is too soon to analyze to a clear conclusion, but I would certainly expect White to win.

This discussion is based on specifics of the particular position, and not on appeal to any general principles. That is almost always the way to go.