1a is correct, 1b is incorrect.
2a) was a catchphrase that was still being used as late as the 1950s, but isn't used any more.
I don't trust word origin stories, but the story is that cigars were once standard carnival prizes, and if (a man!) hit the target, the barker would should "Give this man a cigar!" Or, if he missed by a small amount, "Close, but no cigar!"
Sometimes OUT-OF-DATE slang or catchphrases are used as a kind of joke. Movie lines often become catchphrases. If an old catchphrase is used in the dialog of a new movie that can result in the old catchphrase being revived. But I don't think this has happened with "Give this man a cigar!"
You shouldn't modify a catchphrase or idiom. The phrase is "give this man a cigar" and 2b, 2c, or 2d would be possible, but strange. We can check this very quickly in a rough way with a web search. "Give this man a cigar" (with quotation marks in in the search query": 1,060,000 results. "Give this woman a cigar," 6.