'90s is a contraction of 1990s, or whatever other century's tenth decade is implied. So the apostrophe belongs as an example of a contraction or possessive :)

But your apostrophe is upside down (it became a curved single-open-quote in your version). It’s supposed to be this way:

’90s

Just 90s if it's a reference to someone's age. And the apostrophe for referencing the 90s era is optional outside of formal writing and a dash can also be used e.g.; 90s-music depending on context.

No because it's plural so the apostrophe would come after the S

As bitjunkie said, if you were referring to 90s in the possessive, it would be 90s' since it's a plural noun already - much like parents' mortgages or stores' buildings.

That said, I would probably look at the phrase "90s child" as either (1) a compound noun not needing anything to be possessive (like "ice cream"), or (2) with "90s" as an adjective modifying "child" (like "latchkey kid").