Inability to type in fullscreen is a Flash Player security limitation. Here's what Adobe says about it:
Users cannot enter text in text input fields while in full-screen mode. All keyboard input and key-related ActionScript is disabled while in full-screen mode, with the exception of the keyboard shortcuts that take the viewer out of full-screen mode.
You can read more about it here: http://www.adobe.com/devnet/flashplayer/articles/full_screen_mode.html