It is more naturally translated "Do you have any other questions?".
问题 can also mean problems or issues, not just questions. It's quite commonly used so the answers should reflect that.
Could this also mean "Do you have another question"? (or is 其他 only used with plural nouns?)
I think it could be either. 其他 means "other".
Report it! Maybe DL hasn't saved this alternate answer on its database.
Are 其他人 and 别人 the same?
Usually, yes. It also depends on the context and the sentence.
"have you other questions" not accepted. I still don't understand Duo's obsession with "Do you have" when "have you" works just as well!