No, I don't think it does. It would not be natural to respond with just "Give you?" You might echo the whole phrase - "Give you the book?" - with an appropriate intonation of surprise. Or you might frame it in a question "What do you mean by 'give you'? I can't give you the book. You have to pay for it."

In a daily conversation and storytelling, yes. It's another way of saying: "Should I give you the book?" or "Why would I give you the book?" However, it's best to avoid using this structure in formal writings.

No, it doesn’t. If I understand what you want to say you could use Why should I ?