But don't push too far, that is, don't make this unique thing take the place of the variety of accomplishments that you need to show in order to make yourself look compelling to those who will read your statement. Make sure others get to read the essay before you print out the final copy. This is not just, or even primarily, for proofreading, though this is important. You can miss the mark in an essay like this by being too humble, or by being too arrogant. It is often easier for someone else to pick up this kind of thing than for you to pick it up yourself.
Most schools require two, many require three.
The more detailed and explicit a letter is, the more good it will do you. Indeed, you should offer to give the professors copies of the papers you wrote, to allow them the opportunity to refresh their memories. If one of us went to the school to which you are applying, you can ask that person to write a separate letter. Application deadlines range from late December through mid-February. Ask your referees in November.