Is "De ger inte mig boken" also right?
The machine is clearly pronouncing it "dom"
Yes, that's how de/dem is pronounced.
Would it be correct ''De ger mig ingen boken.''?
No, it would have to be "De ger mig ingen bok", but that means They give me no book rather than They do not give me the book. So that translation is not accepted here.
How would you say, "they won't give me the book?"
I'd probably translate that the same way, assuming you mean that it's currently happening.
why they are not give me the book wrong?
"They are not give me" isn't valid English. It needs to be either "they do not give me" or "they are not giving me".
I was marked wrong for They don't give me the book. Contractions have been accepted before.
Either a bug or a typo, then - I double-checked and it's definitely (supposed to be) accepted.