probably all the smart FlexJS guys are busy or more focused on the dev list so I'll try to answer:
>Does FalconJX do this?
This is a quote coming from the latest FlexJS release announcement  and explains what FalconJX is:
"Apache Flex FalconJX is a next-generation MXML and ActionScript
cross-compiler. It extends the next-generation SWF compiler known as
Falcon. Both are contained in the release package and are used by the
FlexJS package to compile SWFs or cross-compile to HTML/JS/CSS."
So we have to speak about FlexJS.
>Interested in cross compiling existing Flex source to HTML/CSS/JS.
As I got in touch with FlexJS the first time I also thought I could just cross-compile existing Flex code (mx/spark) to HTML/CSS/JS. But that's not the truth. Instead, you have to migrate your code to FlexJS.
That means that you have to remove all flex/flash dependencies from your AS3 code and you have to replace your mx/spark components by FlexJS components.
Depending on your code base that could be more or less work.
So, what remains?
FlexJS places you in the position to develop your apps in a similar way like Flex by using typed AS3 and
MXML and that's great! I perceive that FlexJS is making great progress thanks to the guys who contribute to the project.
I'd like to encourage you to just download and install it and check it out .
Make sure to also check out the impressive MDL (Material design examples) .
I would add to this one information that I recommend you to use FlexJS Nightly 0.8 version where we have a lot of bugfixes and new features.
You can download nightly using Flex installer - right click on combo box with SDK list and choose "Show Dev builds"