Translation:I do not have a girlfriend.
Of course. This is where the multiple possibilities of a translation without any significant road signs can lead. Both Duolingo and your translations are correct. But if we were to use it in some conversation, it would probably be better to choose one or another for the sake of context.
If it's already obvious who you're talking about (e.g. when answering a question whether or not someone is there), simply いません。would do. Otherwise, specifying someone by name or other identifier (e.g. position in a company) would be common. In vague cases you could use a vague identifier, e.g. if there was a man and a woman, and now the woman is gone, you could say 女のほうはいません
That said, "She isn't there" is still a correct translation of this sentence. We really shouldn't have to keep in mind for which lesson a sentence was written.
No, "she is not my girlfriend" would be something else.
There are 2 Japanese verbs that means exist, have, there is..., there are... "Iru" いる for animate objects, and "Aru" ある for inanimate objects.
いる (dictionary form) います (polite) いません (negative polite)
ある (dictionary form) あります (polite) ありません (negative polite)
いません means more like "does not exist" or "is not here" rather than just "not". I'm not sure what the best sentence for your example would be, but maybe something like かのじょではありません.
The sentence is ambiguous. If you specified an object or topic, it would change everything.
Japanese rarely talk about people in the third person though, so it is best to assume かのじょ probably refers not to a specific person, unless otherwise stated.
かれ and かのじょ, あなた, generally aren't used liberally like the are in most western languages.