Good questions...

The design document is really intended to help you guys formulate a development plan. Diving in and just coding isn't a good idea as you'll inevitably end up reworking stuff. If you've got at least some idea of what you're going to do, you can set yourself targets and goals. A to-do list is a great motivator as it's a nice feeling crossing things off the list.

What should the design document contains? I would suggest at a minimum, an outline of your game, information about key gameplay elements and the control schema. You may also want to set yourself some goals and make a little project plan, drop a storyboard in there and some sketched up levels for example. As I said, the document can be an aid to you, so you should put into it the stuff you think will help you.

Will it be published? As with all competition submissions, it will probably be made available for public consumption after the close of the stage.

In terms of whether your game design can change, it has happened in the past... a competitor submitted a design document and then changed their mind part way through, so changing your mind or evolving your design isn't out of the question.