1. What lie did Man 1 tell the king?

2. Which lie did the first man tell the king?
3. What was the first man's lie for the king?
4. What was the first man's lie to the king?
5. Will you tell me the first man's lie?
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They are several questions which mean the same thing? Do they need any corrections?

All are grammatical, assuming the liar in #1 is designated as Man 1, and it is used as a proper name. #3 could be confusing. Sometimes we refer to a spokesman FOR the king as telling lies on the king's behalf, or for the king.

Do you mean #4 is the beter of the two?

Yes, #4 is better than #3 as long as you mean that the man was telling the king a lie.

No, all the questions you've listed are asking about the same thing, which is the lie told by the first man to the king. They are phrased slightly differently, but they all inquire about the same piece of information.