To me this English sentence would likely be used in one of two situations: there is a physical stack of newspapers present, and the hearer is being invited to read one of them or one is conveying a subtle jab at the hearer's current events savvy, as "Read the newspaper" is so much more natural as a general request to read even any unspecified newspaper (it's kind of like "I ride the bus"; there's not really a specific bus).
Would the Turkish sentence be used similarly?
The suggested translation above being, "Read a newspaper," it sounds like a technical glitch. These happen sometimes. If it happens again, take a screenshot and file a bug report: https://support.duolingo.com/hc/en-us/articles/204728264-How-do-I-report-a-bug-
You mean a newspaper. We should assume "a newspaper" if the object isn't in the accusative. Accusative ends in: ı, i, u, ü, yı, yi, yu, yü, nı, ni, nu, nü. For gazete, it would be gazeteyi.
Note that this is valid only for the object of the sentence. A subject can never be in the accusative. So if you see a sentence like "Gazete orada". You have to decide whether you should use "the" or "a" judging by the context.
In this sentence would "gazeteyi" be used to refer to the local newspaper of the place where one currently is (or if there happens to be a physical, printed paper that is being referenced) and "gazete" if there are a number of newspapers available, and the request is to go read any one of them?
It seems to me that in English we often say "Read the newspaper" even if it isn't clear which newspaper is actually meant (which is very frequently the case, given that so many people read them online).
True. In English, it's almost a fixed expression. In Turkish, it just has to do with whether or not the listener is aware of which newspaper you're talking about. Let's say the newspaper has been lying on the table, and there was some interesting news in it. I could ask "Gazeteyi okudun mu?" - Did you read the newspaper? You know which one I'm talking about. I don't know if other newspapers wrote this news, but the one on the table has it. Did you read that one?
But if someone asked me what I'm doing right now, I could just say "Gazete okuyorum." I mean, you don't need to know which specific newspaper I'm reading. I'm just informing you that I'm on the couch, reading a newspaper. You could, though, ask afterwards "Gazeteyi nereden aldın?" - Where did you buy the newspaper at? Because there aren't any kiosks nearby to buy newspapers at. Or something like that.