It wouldn't sound completely wrong, but it's not that commonly used. Here, the indefinite article is not ommited because someone is hearing a particular noise,and it might be only one. Ακούω θόρυβο makes it a bit more general. Ακούω θορύβους (I hear noises) is a bit more commonly used in Greek. ^.^

ִIt means that the sound is ongoing, when you omit the article like that. It amounts to I hear a fuss/noises - examples. Ακούω θόρυβο από το πάρτυ/ακούω θόρυβο από το ψυγείο. I'd say your suggested answer sound be marked as wrong.

For me it should be accepted, I can see no difference at all.

Those are different cases of the same word. The word is conjugated like that:
Singular
Nominative | ο θόρυβος
Genitive | του θορύβου
Accusative | τον θόρυβο
Vocative | θόρυβε

Plural
Nominative | οι θόρυβοι
Genitive | των θορύβων
Accusative | τους θορύβους
Vocative | θόρυβοι

Well, according to the official grammar rules, the final -ν is dropped when the next word begins with a β, δ, γ, φ, θ, χ, μ, ν, λ, ρ, σ, or ζ. But we do accept both spellings. ^.^

Edit: But, not for maculine nouns. The final -ν is not dropped, so that there is no confusion between the masculine and the neuter pronouns and articles. D_. pointed that out right below, I not-so-gracefully ignored it somehow. ._.

There are a few sentences like that in the course, butt he ssue will be fixed with the release of the new tree.