Translation:I believe my classmates have the same question.
Certainly it would not be "abbiamo" as this is present indicative for "we are" and the sentence would not be correct obviously. It could be "hanno" for present indicative of "they are" and I believe this would be correct and that the sentence would make sense. "Abbiano" though expresses the subjunctive mood as in '"they may have" the same question' in which an uncertainty is being expressed and this probably works best in terms of what the sentence is meaning to convey. There is a great explanation of the subjunctive at this link provided by adamyoung97 - https://www.duolingo.com/comment/8783716/Italian-Subjunctive-Guide
A question is not quite the same as a request. A question would most likely be an enquiry, whereas a request is an attempt to get something from someone. E.g. In class a question might be "What is the capital city of England?" compared with a request "May I leave the room, please?"
"Companions simply would not be used by a native English speaker (except, perhaps, ironically). "Classmates" is normal and as an English native I would expect to (and did) translate the Italian thus. "That" is technically correct and I try to use it wherever it introduces a subordinate clause but it can sound a little too correct and unnatural and is often omitted; after "I think" and "I believe" for example.