As far as I can tell the issue is in facebook code.
It installed a key listener on the comment field (for whatever sinister purpose it needs it I do not know, please make your own conspiracy theories ;)) Transliterator sends backspace keys to the input field to replace original input with mapped characters. Facebook ignores the backspace keypress events, and echoes back what it has logged after transliterator is done.
re: F2 - There is no global shortcut registry in the world, I cannot tell Google that F2 is already reserved for something else. So - change F2 to some other key or use the menu.
Come to think of it, Google Sheets is a perfect candidate for full-window transliteration mode ( Shift-F2 by default ) - this turns on transliteration for all cells instead of just one at a time.