I suspect flash is generally used to play sounds from chat messages - the https man-in-the-middle detection is heavily sampled, as referenced in https://www.linshunghuang.com/papers/mitm.pdf.
[I work at FB, but not on sounds or directly on https man-in-the-middle detection.]
Nope, without flash you still get the chat sound messages. I've no flash on my system and the only thing that's different on facebook is that I can't watch user-uploaded videos. Only their mobile site supports HTML5 last I checked.
It is still possible that they use flash as default audio source and fallback to HTML audio if flash is unavailable. Although of course it would be better if they could get rid of the flash altogether.
[I work at FB, but not on sounds or directly on https man-in-the-middle detection.]