Learn English – a thousandS years tradition

In English, every number that is not 1 is considered plural. The correct sentence is the first you wrote.

Your password expires in 0 days.

A 16-year-old girl:
1) "16" and "year" are linked by the hyphen to create a single term ("16-year") that modifies "old."
2) "16-year" and "old" are linked by the hyphen to create a single term ("16-year-old") that modifies "girl."

She is 16 years old:
1) "old" is a predicate adjective for "girl." (as in "She is old.") As a single-word adjective for "she," it is not joined with a hyphen to the previous adjective, "years."
2) "years" modifies "old," and "16" modifies "year. "16" and "years" are not acting as a single adjective for "old," so they, too, do not need a hyphen.

The easy way:
If the adjective string serves as a single modifier before the noun, it needs a hyphen (or hyphens, as in this case). If it is after the noun, it doesn't need a hyphen (or hyphens).

This is the same as "3-day weekend" and "The weekend has 3 days." In the first case, "3" and "day" are linked to create a single term that modifies "weekend." In the second case, "days" is the object of "has," and "3" is a modifier for "days."