Using .NET (Windows), you can embed an ActiveX Flash player into a Windows form, load a swf file and set variables or invoke functions with single method calls from C# (I imagine because of the presence of a Flash OCX).
That said, I would like to do the same thing in MONO (Linux): how could this be possible?
Is it possible to "embed" an HTML page, using some (system controlled) prefered renderer (like firefox)? In that case it might be possible to get a flash in there and control it through javascript... It does sound a bit far-fetched though.
If you do manage to get the flash in there, you might be able to communicate with it through a TCP/IP connection from the flash to your program.
Flash takes the form of an ActiveX (i.e. COM) DLL on Windows because that's the preferred extensbility platform for Internet Explorer. In other browsers, and on other platforms, Flash takes a different form, using the browser plugin API I believe. Since ActiveX is a no-go on Linux, perhaps it might be possible to implement a plugin host of some sort that can make the Flash plugin think it's running inside a browser. This would probably be "non-trivial" as they say, but theoretically possible.