A few answers based on how I work, may not be for you, just my two cents on the subject.
1) What is best way to collect and organize a bunch of note's and idea's?
A. This can vary but I have found Articy Draft to be a great tool to store your thoughts regarding characters, story, even map ideas, etc. http://www.nevigo.com/en/articydraft/overview/
Prior to this I would use old school pencil + paper in a large notepad and write down any thoughts that came to mind. I would carry this over to a word document which contained a more final version of the story, this step has been replaced with Articy Draft, and I find myself brainstorming using this software now.
*Edit* The Articy Draft tool has student and education pricing as well I believe. Thought this was work adding since it can appear as expensive for someone starting out.
2) How should I put the story together once I have enough notes and idea's for the game?
A. This is the part which requires some dedication, it is part creative thinking, and part sit down and just start writing. Set a block of time aside for yourself which is dedicated to writing and use that time to brainstorm and think about how your character(s) will be progressing from event A to event B in your story. Part of this is creative in the sense that you will get some 'cha-ching!' moments when you connect the dots spontaneously with an idea that just jumps to mind. The other half is actually theorizing what is missing from this chapter / story, what obstacles do my characters need, does this make sense. I would reccomend watching this clip regarding creativity from John Cleese back in the 90's. It is old but it is brilliant, and it is also explains why setting a certain amount of time aside for this sort of thing is importnant. http://www.youtube.com/watch?v=tmY4-RMB0YY
3) Should I name the game after the direction the story takes and the story is all written and finalized. Or should I figure out a name then have the story develop around the name I have picked out?
A. I have trouble naming things as I am working through them because things about the story change, or how I interpret the meaning of the story changes, I usually end up naming things after they are close to completion. (A lot of my work stays 'open' because I hate finalizing things completely ). Names to me are typically a reflection about what the story meaning is, or a summary of their journey, or even named after the character if the story is heavily focused through the viewpoint of that one character.
4) Should the storyboard be included inside the design document for the game. Or should it be a separate document in itself?
A. Story boarding is the act of connecting different story panels together, traditionally used in comics, and animation. It is unlikely you are going to storyboard the entire story, otherwise you would have a lot of panels, but it is not unrealistic for your to storyboard the climatic events in your story, or choices for the dialogue in your story. These junctions can be story boarded in your game design document as more detailed examples of the event in that part of the story. General rule, there is no size constraint for your design document, if you are willing to put more info into the design to clarify things for those on the team then definitely add more information, as long as it does not confuse those who are reading it.
5) Where do I begin after all my other questions have been answered?
A. If your goal is create the story for the game, then begin writing. Even if you are not 100% sure where your story is going, or what the story might be, set some time aside which is dedicated to writing and all things creative and just start. Writing down the story to events that happen at the end or the middle of the story is fine, you don't need to write it from beginning to end, just write what you have, and start somewhere. The important thing is that you START, not how or where you start.