I agree it works better in indefinite form in English. However in Danish it has to be in definite form.
In English "the nature" would more likely be referring to the nature at some specific place, whereas simply "nature" is more like the abstract concept of nature in general. In danish this concept would be "naturen" not just "natur".
In that case the correct answer should be given as 'nature', not 'the nature', otherwise it is misleading as to how we should actually use the word. If we learn that 'naturen' means 'the nature' then if we ever want to say 'nature' we will think it should just be 'natur', which would be wrong.
I think it's because of the definite form of "natur" and the present continuous (for once it's used! lol) of "leve" that 'the' and 'living' is included in the best translation. It may be because a literal translation may be required whenever possible, but I'm not entirely sure, it's my best guess. Your version without "the" and substituting "living" with "alive" is also accepted as a translation :) However, "environment" is "miljø" and not correct.