Without having an adaptor, an American Turbo Grafx or Turbo Duo can only play American Hu Cards. Likewise, a Japanese PC Engine/Duo can only play Japanese Hu Cards without an adaptor.
There were three different CD formats for the PC Engine/Turbo Grafx's cd add on. Regular CDs, Super CDs, and Arcade CDs. The Turbo Duo can play both regular and Super CDs if they're from America or Japan. The Turbo CD add on for the Turbo Grafx needs a specific Hu Card called a System Card to play CDs. The system card 2.0 lets you play regular CDs, and the Super System Card 3.0 lets you play Super or Regular CDs.
As for Arcade CDs, in Japan only a 3'rd memory upgrade came out called the Arcade Card. It let you play Arcade CDs. There were actually two Arcade Cards produced. The Arcade Card Pro, and the Arcade Card Duo. The Arcade Card Pro worked when you had a PC Engine with the CD add on, and the Arcade Card Duo worked on the Japanese Duo.
So, if you want to play an Arcade CD on your American Turbo Duo, you will have to buy a convertor, and an Arcade Card Duo.
As for PC FX cds, the PC FX was more like a Playstation, as opposed to a Super Nintendo. It is an entirely seperate console, and is not backwards compatable at all. I hope that clears some things up for you.
~Retekin