ah ha! Yes it does fold down! It has a hard back and you can put stuff on it. I say this with a laugh, because I probably took 20-30 flights in F before figuring that out myself. I was pretty stoked when I found this out just last year.

Although maybe we're not talking about the same thing. What I'm talking about on the F seat is the extra cushion part, that is part of the main seat, but the fabric itself is separate from the wider part of the seat. You can pull that narrow section down, and on the back there is a hard surface. Personally I wish they had some type of cup-holder functionality there.