In English, the following sentences mean two very different things:
His dog is not eating meat. His dog does not eat meat.
However, it seems that in Swedish this sentence is used for both meanings. If I were reading a book in Swedish and came across the sentence, "Hans hund äter inte kött," am I to understand that his dog is a vegetarian, or just that his dog is not actively eating meat? Does the Swedish language rely on context clues for the distinction?
It depends on context clues.
(I am a native Swedish speaker) In this case I read "Hans hund äter inte kött" as "His dog does not eat meat", though I suppose it could be taken the other way.
If I want to be really clear that the dog is not eating the meat that's currently in his food bowl, I might say "Hans hund äter inte köttet." ("His dog is not eating the meat.")
In Norwegian, both forms "min hund" and "hunden min" are accepted. "Hunden min" is the most common form, while "min hund" is stressed on "min" and emphasises the fact that the dog we are talking about is mine. Is this true for Swedish too, or is "min hund" the only correct form (or the most common)?
So the negation of the object versus the negation of the predicate matters in Swedish? I would contend that the difference is so subtle that it doesn't really matter, at least to a speaker of American English. The difference is in the emphasis alone. The underlying meaning remains the same. Unless the meaning of this particular sentence is limited to the immediate present, as in "His dog is not eating meat right now," but it might eat meat at some other time. In which case, "His dog does not eat meat" is a false translation, because "His dog does not eat meat" actually means that the dog never eats meat. Ever, in its entire life.