"Could you" is more like suggesting if the person can or cannot do something. "Would you" is more like suggesting if the person wants to do something assuming that they could do it in the first place. "Will you" i feel is a very direct yes or no question which is basically the same as "would you". This is from the top of my head as a native English speaker :) (Feel free to critique)

"Could you" is the most formal of the three and used for example: In a restaurant: " Could you please bring me a water" Could we please have some bread" "Would you" is used similar to how "Could you" and more formal. "Will you" is more informal and used more with family and friends. To your child: "Will you please clean up your room" "Will you put your toys in your room". It's more of a demand and less of a question.

Although, as you have said, all three of these are quite similar, I would explain the difference as follows: Could you implies that there might be a reason that the person you are asking can't do what your asking, not that they don't want to but that they perhaps just can't. Would you implies that it is at their option (the person's you are asking) and you are simply asking if they would or not. Will you is the most definite of the three, you are under the assumption that the person could and would do the thing you are asking them to and politely telling them to do so. For the other person to object after you have asked in this way may seem a little rude as compared to the previous two (could/would). -Eric