Why is it that in this sentence, it's grøn kjole; but in the sentence with the woman having a green dress, it's grønne kjole?
If the sentence was "Kvinden har en grønne kjole" then it would be completely wrong. However, if the woman has the green dress, then it would be "Kvinden har den grønne kjole". In Danish, when there is a noun between the definite article (den/det/de) and the noun, then the adjective takes the "e-form" (also true if a possessive pronoun is before the adjective + noun). For example:
Huset er smukt = The house is beautiful
Det/Hans/Mit/Anjas smukke hus = The/His/My/Anja's beautiful house
Et smukt hus = A beautiful house
Graou´en, is how it sounds to me. But you have to say it together quickly. Like grou'n.