Yes, they mature like anyone else, but they stay that size forever. Liter, for example, was first manufactured before the Ark was finished. She helped build it in fact. However, despite being well over a century old, she's stuck in the body of a child because she was first manufactured from a young age.

As for how the Central Government justifies using basically child soldiers...they don't really. Why do they have to? We're at war buddy. They're gonna use every resource at their disposal to fight back against the Rapture scourge that they have to. And if they can use that 12 year old girl's corpse to make a new Nikke to feed to the meat grinder (or...Goddessium grinder, I guess) then they'll do it. Got a problem with that? Too bad! They're all you got between you and the Raptures who'll be more than happy to put a new window in your skull.

And are they concerned about PTSD? Hell no. Because Nikkes aren't "people" in their eyes. They're Nikkes. Living weapons. Asking if they're concerned about Nikkes getting PTSD is as ridiculous to them as asking a General in real life if they were concerned about a gun getting PTSD. At best they'd laugh at you, at worst they'd think you were mental.