This is not a normal English grammar structure.
The rules of Jeopardy require that contestants answer the clue in the form of a question.
I tend to think of it this way... it's a backwards game show. Most game shows require that the host ask a question:
Who was the first president of the United States of America?
And contestants provide the answer:
George Washington.
With Jeopardy, it's the other way around. The host is answering the question while the contestants are asking it...
So, if the clue is:
This man was the first president of the United States of America.
The rules require the response be
Who is George Washington?
If they fail to do this, even if they're technically correct, they lose points.
So, if a contestant answered:
George Washington.
They would be wrong by the rules of the game.
As to their hesitant tone, sometimes they're not completely sure of the answer, so they're guessing.
What CDO is giving you is typical uses, not restrictive definitions. What precisely (or vaguely) is meant by these utterances is more dependent on the discourse situation than on any specific meaning either of them expresses.
For instance, one of my colleagues may ask me tomorrow "What was the concert like?" I know that she doesn't know anything about the music of Koji Kondo and that what she's really trying to convey is that she remembers my mentioning last week, with some bemusement, that I was taking my son to a performance of music from the Zelda games and she hopes it wasn't too tedious for me. I will reassure her that "We had a really good time".
On the other hand, my son's friends are mostly musicians, and if one of them asks him "How was the concert?" he will probably answer something like "The first trumpet seemed to have chops problems, he had a lot of high entrances and was usually a quarter-tone flat. But the arrangements were very cool, a lot of Debussy in it, and the video synch-up was kick-ass."
Your own phrase, "giving freedom for the other person to choose the content of her answer", expresses the situation admirably. Neither I nor my son pay any attention to how the questions are worded: he and I, and our interlocutors, address our audiences and our social situations.
Best Answer
In fact, you should always use "please" when you ask someone a permission a favor or something else.
So in your question : "Could I please send you a letter ?" as it requests an authorization from the recipient you may put here your please.
(However, I'd start with may instead of could, execept if your question is following ou implying a previous condition...)