I am curious why there is an article "a" in front of "living"
What would be the difference between "for living"
And is "living" in this case a noun or a verb?
The indefinate article is always found with such an expression.
The original meaning is lost if for is removed.
What do you do for living X
Living appears to be a noun due to it having an article and being an object of a prepostion (preposition objects are often noun phrases).
This (below) information confirms that it is, in fact, considered a noun.
1 the money that you earn from your job:
What do you do for a living (= What is your job)?
I mean, I don't like my job but at least it's a living (= a way of earning money).
You can make a good living (= earn a lot of money) in sales if you have the right attitude.
See also livelihood.
Cambridge Dictionaries Online - Cambridge University Press
I am not a teacher.