Questions & Answers about Cykelhjelmen er min, så den tager jeg.
How would I ask "Whose bicycle helmet is this?" to get this sentence as an answer?
You would use the interrogative possessive pronoun hvis (whose). Because hvis does not change based on gender or number, the question is simply: Hvis cykelhjelm er det? (Whose bicycle helmet is it?).
Why does the second clause say den tager jeg instead of jeg tager den?
The object den (it) is pulled to the front of the clause for emphasis. Because Danish strictly follows the V2 rule (the verb must be the second idea in a main clause), the verb tager stays in the second position, pushing the subject jeg to the third position. It gives the subtle meaning: "...so that is what I will take."
Why is it min instead of mit or mine?
The possessive pronoun must match the gender and number of the noun it belongs to. Cykelhjelm is a singular common gender noun (an en word, which we can tell from the definite form cykelhjelmen), so it requires the common gender form min.
The English translation says "I will take it," but tager is present tense. Is that normal?
Yes, Danish very frequently uses the present tense to express future actions, especially when the decision is immediate, natural, or certain. Using a future construction like vil tage here would sound unnecessarily formal or like a rigid promise.