"¿Cuántos hombres estudian aquí?"
Translation:How many men study here?
Forming a proper question in English is complicated. In such a question that includes a question word, you only need an auxiliary verb if the thing you ask about is not the subject.
Asking about the subject:
- How many people work here?
- Who slept in my bed?
- What bothers you?
Asking about the object; the subject is marked in italics:
- How many people do you employ?
- Who did John sleep with?
- What do you do?
Study is a full verb so "do" isn't required (like in "do drugs") but it could be inserted if you wanted to emphasize something like the reality of their studies.
My answers 'how many men do study here?'. How could it be marked incorrect?
It is not exactly wrong, but it's not the most common way to form this question. Usually, if you ask about the subject, you just use the full verb:
- How many men study here? - "Men" is the subject.
- What do the men study here? - "Men" is the subject; "what" is the direct object.
- Where do the men study? - "Men" is the subject; "where" is an object.