In American English, it is "What do you do for a living?"

I would say that "What do you do for a living?" is correct. If you remove the "a" it only sounds strange to me.

What do you do for A living?

"What do you do for A living" is correct! "What do you do for living" sounds a bit funny but I'm sure native speakers would understand!

What do you do for a job? What do you for for a living? What do you do for a crust? These all ask about the same thing: what kind of work you do, or what your profession is. "What do you do for living" sounds awkward. It also sounds like a different question.