I was wondering whether it had that connotation like in French. I would guess if it's not accepted that means the language doesn't imply that. Google translate gives "fiica mea" for "my daughter", so whereas in French there isn't a word to differentiate between girl and daughter, but only the possessive adjective, it seems like in Romanian like in English there is a different word so we're expected to translate "fata mea" directly to "my girl".
Because the definite article is already covered by the numeral.
This is generally true for cases in which you switch the noun-adjective order - the noun loses the definite article which is attached to the adjective:
The beautiful girl is there. -- Fata frumoasă e acolo. / Frumoasa fată e acolo.
In the sentence at hand, you have to do this because you can't place the numeral after the noun.
It adds emphasis on the adjective. These noun-adj inversions are very common in literature - they are perceived as artistic. Plus, from a technical point of view, it's a great way to control rhythm or rhyme in a poem:
A fost odată ca-n povești,
A fost ca niciodată,
Din rude mari împărătești,
O preafrumoasă fată.
(Mihai Eminescu - Luceafărul)
You can see a switch, but with the indefinite article; also "preafrumoasă" means "very beautiful".
Disclaimer: I'm just a native speaker with an active interest, I have no serious formal training in linguistics. This exact topic seems very complex to me; besides what's described above there are other situations in which the noun-adj order is changed and I probably couldn't exhaustively cover them all.