'Don't have' translates as 'no tengo' .
eg I don't have time / I don't have any brothers or sisters
so that isn't possible here.
In your sentence about the exam, you can answer either:
I didn't pass = past simple
or
I haven't passed = present perfect
These are similar in meaning, and would depend on the context. If you're not sure whether to use the past or the present perfect, I'd advise you to use the past. It's more likely to be correct, especially in US English.
I hope that helps.