Do put quotation marks on nonfiction stories?
yes of course you do
in any story you can put quotations in
Yes, the names of short stories should go in quotation marks. The general rule is that lengthier, stand-alone works, such as novels and plays, should be underlined or italicized, while shorter works, such as short stories (what I'm assuming you're referring to) and most poems should be placed in quotation marks.
Yes, the titles of poems are put in quotation marks. The only exceptions are epics, which should be italicized or underlined. The general rule is simple: works that are generally published on their own ("stand-alone" works) are italicized or underlined, while works that normally appear as part of a collection (such as poems and short stories) are put in quotation marks.