"American girls are more beautiful."
Translation:Amerikan kızları daha güzel.
It is a nominalised adjective actually. You should be able to use "Amerikan kızlar", too. However, Turkish has a nominalisation process which may transform adjectives into nouns, just like English's non-distinction between nouns and verbs.
English A three-hour walk ('walk' is a noun here) He walked for three hours. ('walk' is a verb here)
Turkish Güzel bir kadın ('güzel' is an adjective here) Bugün güzelleri gördüm ('güzel' is a noun here, with the plural ending -lAr and accusative case)
So, the answer to the second question is hidden above: it might be the case for ALL adjectives in Turkish, depending on the discourse and syntactic needs. Hope it helps.
Surely the -I ending in "kizlari" is wrong ? It's used correctly after a preceding noun, for example "the teacher's girls" would be "ogretmenin kizlari" because "ogretmen" is a noun. But "Amerikan" is an adjective. "Amerika'nin kizlari" (= "America's girls") would be correct though.