«a girl» and «the girl» not accepted. Why «my girl» is what I must write? There's no «моя» in the question.
If the girl is девочка and the woman is девушка,Why is the woman likes to cook at my house wrong?
Because 'woman' is женщина? Those words don't have exact correspondences (Russian has девочка — девушка — женщина and English has girl — woman, so девушка may be hard to translate, but it's usually translated with the word 'girl').
Except "lady" can refer to any woman, and if you use a more precise "young lady", it will also mark you as an old person.
Is любит готовить translatable also as "loves cooking" or "loves to cook", or is it only "likes"?
I have limited knowledge, but thw verb "любить" generally means like with objects and love with people.
"Lady" is not age-specific in English, while 'девушка' is. If anything, "девушка"="young lady" in English, except I cannot imagine a man below a retirement age saying "young lady" - there is something intrinsically "old geezer" about this construction, when uttered by an American man.