How do I control the play/pause of a video with a key press rather than a mouse click

Flash Player is a language runtime, not a video player.  The ability to launch a video using a Flash-based video player is going to be dependent on whether or not the author of that video player listens for keystrokes.  Spacebar is pretty common, as long as the video player has focus, but again, it's entirely dependent on how the video player you're using was written.

If you can just download the source video, you might find using an actual video player instead of Flash Player to be more useful in this regard.  VLC will typically play .flv videos.