Translation:The teacher is handing out the documents to the students.
No, but "distribue" translates as "is distributing". If you write "is distributing" then "among" is better than "to" here. "To hand out" seems more physical than "to distribute" so implying the passing of documents by hand. In this context it's okay, but distribute is not wrong.
Add one in the "something else" section. You can have both boxes ticked. Besides, I think it's better they get an explanation than just a ticked box the majority of the time. Remind them of their lack of internal consistency and their inability to tell us what they want and why, because there are so few "hints" these days.