Is first lady a proper noun?
It depends on the use of those words. See explanation.
It is a proper noun when you mean one exact person, then you should use capital letter:
First Lady of the United States - Michelle Obama.
But if you refer not to one person, but to the idea of the first lady without mentioning a name then you treat it as a common noun and you do not have to write it with captal letters.
For example: Should the first lady earn a salary? In this sentence you think about any person not an actual one.