How many articles should go in "Merry Christmas and Happy New Year!"?

There's a difference between a complete, grammatically-correct sentence and a greeting.

If I was writing a complete sentence, I would write "I wish you a merry Christmas and a happy New Year." But if I was just greeting someone, I'd say, "Merry Christmas!", not "A Merry Christmas".

It's like when you write a title or a headline, you often leave out words, especially articles and versions of the verb "to be". A newspaper headline will say something like "Stock Market Up", not "The Stock Market is Up".

I suspect people write "Merry Christmas and a Happy New Year", i.e. no article on the first but including an article on the second, because they're starting out writing a headline type of construction but when they get to the middle it's getting kind of long and they put in the article as if it was a complete sentence through force of habit. I think it's inconsistent and I wouldn't do it. I'd generally write "Merry Christmas and Happy New Year!", or maybe possibly "A Merry Christmas and a Happy New Year."

Of course, in these politically-correct times, the correct greeting is "Happy Unspecified Holiday to you, unless the very idea of holidays offends you, in which case please pretend I said nothing and walked past you in silence".

Both seem fine. It is more of thing of conventions rather than rules.

I don't think any of the solutions you posted are incorrect.

Generally sentences become more formal and elaborate, when you use more words. So despite all of your sentences being correct, the most formal one would, in my eyes be

"I wish you a merry Christmas and a happy new year".

Of course you can arbitrarily add more formality by making it longer:

"On this day, celebrated throughout the world as a day of peace, I wish you and your family a merry Christmas and a happy new year".