Translation:The woman eats apples, if she is a vegetarian.
Duo never lies to me. If you're a woman and eat apples, you must be a vegetarian.
When is hvis 'whose' and when is it 'if?' Why not 'Kvinden spiser æbler, om hun er vegetar?'
"Hvis" is "whoose" if it's an interrogative pronoun ("hvis æble er det?"/"whoose apple is it?"), and it would be "if" if it's a conjunction like above. So that's Danish to English.
When translating English "if" to Danish, however, it can be translated to either "om" or "hvis", but they're not synonymous. Here "hvis" is used for logical implication ("jeg bliver våd hvis det regner"/"I get wet if it rains") whereas "om" is used to express uncertainty ("jeg ved ikke om han kommer"/"I don't know if he will come").
Would it be correct to say: "jeg ved ikke HVIS han kommer"/"I don't know if he will come"
As for your second question: om is 'if' as in 'whether' "(I don't know if/whether this will happen'", whereas hvis is 'if' as in 'supposing that' ("if it happens, we will respond").
if it translates 'she is a vegetarian' isn't it supposed to be 'hun er en vegetar'?
Adding 'en' would probably be acceptable (I'm a native speaker of Dutch, not Danish). Elsewhere in the course 'he is a member of...' is translated as 'han er medlem af', with 'han er et medlem af' accepted as an alternative. I think the indeterminate article (en, et) is usually not used when the noun describes what somebody does for a living or what they believe in: 'han er læge' (doctor), 'hun er minister', 'hun er socialist'. It works like that in both Dutch and German, and I strongly suspect in Danish as well. My Google search seems to confirm this.