1) When a video is played by the user (user clicks play) make the video playback window go into full-screen mode (not full-browser). You will need to tie this to a user action event (such as clicking play) in order to satisfy any security restrictions in certain browsers.
2) Add an event handler to fast forward to a specific time in the video when: the video is inside a specific time interval (eg. between 5:32 and 6:01 of playback) in the video AND a click is registered (in full screen mode).
3) Return to normal viewing mode (from full screen) when the video is completed or paused (see below).
4) Pause playback when any key on the keyboard is pressed.
This project needs to result in a fully functional demo of the above behavior and work flawlessly on modern desktop (IE8+, Chrome, Firefox, Safari) and mobile (Android, Chrome, Safari) browsers. I will take your example and use it to integrate into an actual use-case which requires a user's full attention during viewing of video material and will shift to different parts of the video based on user feedback (mouse click) at certain points in the video.