Well, "I do not have an account in the bank" leaves the possibility of having an account in a different bank, so I don't think your answer is right.
It depends on how the Danes use it. To me 'Jeg har ikke en konto i banken' usually would mean that you don't have an account at the bank as a general thing. So then bank account should be correct in my view. I reported it. Let the native Danish speakers decide. :)
In the bank, sounds unnatural to me.
They all seem correct to me. I think, 'I don't have a bank account,' is the most common in the UK.