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As far as I know, linguists determine grammaticality by judgement tests. Native speakers are presented with several types of utterances and, based on their intuition, they can judge each of these utterances as being grammatical or ungrammatical.
But languages like Latin and Gothic, for example, are not spoken natively by anyone anymore. So, does it make sense to talk about grammaticality for dead languages?
That may depend on your definition. We Latinists trust in corpus research and what Roman grammarians said themselves about what they found acceptable. Of course we have less certainty than with living languages, but overall it isn't really a fundamental problem.
It could even be said that corpus research is more reliable than judgement tests, because people may say they would "never write" certain things that they do in fact write on occasion. (Keep in mind that the Latin we know is a literary language: we don't make many claims about spoken Latin. Grammaticality differs between spoken and written English too, at least for most people.) I have heard many Dutchmen claim they pronounce the n of the infinitive (lopen) and plural nouns (apen) who wouldn't accept that they did not in fact pronounce it. Another example is the famous linguistics student who said in class, "that is not true, Professor: I don't say don't!", then cowered in shame.
One interesting source of information comes from what old prescriptive grammarians found to gripe about.
Every now and then you'll see someone write something like "It's very bad to use the form X. You should only ever use this other form Y." What that tells us, in an indirect way, is that some people had started using X. (After all, why would you complain about a "mistake" unless some people around you were actually making it?) In other words: for at least some speakers of the language, we can conclude that X had become grammatical.
So in these cases, we can learn about how people actually talked, and not just how they believed they should talk. But this sort of thing is pretty rare. Usually, yeah, it's just corpus work on the written language, like Cerberus describes.
No, I'm afraid you are wrong. In generative syntax, there are two distinct concepts, acceptability (=the native speaker's intuition) and grammaticality (=observing or violating rules of grammar created by the linguist). You can read more about it in any intro textbook on generative syntax (Haegeman 1994, Adger 2003 etc.).
No, It doesn't make sense for exactly the reasons you posited.
Grammaticality judgments require a native speaker. It's safe to say that anything found in the corpus of a dead language is grammatical but there is no way of judging the grammaticality of anything not in the corpus.
