Wouldn't ¿Crees que has aprobado el examen? be "Do you think you have passed the exam?" I don't know the grammatical tense that applies, but on the face of it, it seems the "have" should be represented in the English.
Also, does Spanish have an equivalent of "do?" I think i read that it does not. In English it is often ommitted, so it's probably ok to not always include it in translations.
The preterite is often used in English when we use the present perfect in Spanish, which is the case of this test-question. For the use of the present perfect in English there is generally an extra nuance, and there is no more context in this text, so as a default, the preterite is used.
With regards to the word for "do" in questions in Spanish, you're right, we do not have any word reflecting that. When we ask we simple conjugate the verb straight away with no extra words (do..._?, did... ?, will... ?) As we have two interrogation signs "¿...?", you would know that that is a question.
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