Will 2x11-speed front derailleur work on 2x9 system?
The short answer is yes, it should work.  I like to refer to Sheldon Brown's site because it tends to be clear and fairly technology-neutral.  It has this to say:

Pretty much any front derailer will work with any modern chain system.
It is true that the cages have gotten slightly narrower to match the narrower chains used on systems with more rear sprockets, but this is rarely a problem in practice.