I don't think it's a fixed phrase. It's just a construction «нет» + Genitive (used to express absence) combined with the adverb «уже́» 'already'.
Hmmm, but the fact that some phrase is fixed is not determined by the English translation.
E.g. I can’t think of a phrase where the English ‘black olive’ can’t be translated as «маслина» in Russian text. Does that make English ‘black olive’ a fixed phrase? I’d say no. It’s just a combination of the adjective ‘black’ + the noun ‘olive’ in their normal meanings. The fact that Russian has separate words for green olives and black olives doesn’t make ‘black olive’ a fixed phrase in English, because Russian doesn’t determine what constitutes a fixed phrase in English.
Similarly, the fact that English has a separate word for ‘ещё не’ doesn’t make «ещё не» a fixed phrase in Russian, because English doesn’t determine what constitutes a fixed phrase in Russian. It’s just a combination of «ещё» and «не» in their normal meanings.
Spanish grammar seems so far to be so, so much similar to Russian grammar than to English. If only there was a Spanish - Russian course :P
For example "уже" works quite like "ya" in Spanish, while "нет" works like "no hay". And, unsurprisingly, "уже нет" seems to map perfectly to Spanish "ya no hay".
While I agree about the status of being a fixed phrase being independent of phrases in other languages, I wouldn't argue that the combination of the concepts conveyed by уже and нет is "there isn't ... anymore". The only thing I can really think of is running out of something unexpectedly soon, which is definitely not what's conveyed by the translation here.
The latter. Probably the school was closed because all the young people left the village and there's no one to teach, or something.
Another example of the pervasiveness of North American English grammar library software.
In British English, any more refers to both quantity and time. In American English, anymore is limited to an aspect of time.
Just to add to the confusion many Americans adhere to the British practice of using any more to refer to both while other Americans misuse anymore and use it to refer to both.
Anymore is an American usage of any more that limits any more to an aspect of time.
American usage: We don't have any more books. (quantity). We don't have books anymore.(time).
The British do not use anymore to make that distinction. Not every American is aware of the distinction. Both the British and the Americans make little attempt to avoid running the words together in ordinary conversation. As a result, it usually sounds like anymore no matter what is intended.
This is one of those moments when Russia's love of shortening sentences is officially a detriment. I had no clue what the sentence could mean because, without context, it was literally "no, school already not"... Not what? Not open? Not here? Am I not attending or am I talking about someone else not attending? No lie, I failed this one on purpose to see the answer, and it still doesn't make sense to me without a location added to the sentence.
I think that you are confusing не with нет. не is the one that works as "not", but "нет" is most of the time translated as "no" and thus can stand on it's own. Like you say, "школы уже не" would totally look like an incomplete sentence, but this one uses нет. Also, remember that "нет" with genitive has a special meaning of "не + verb-to-be-in-the-present" (just like не было is used for past sentences, but быть is ommited in the present).
I think that is correct, but emphasis changes.
"Школы уже нет" places emphasis in "уже нет". If you were talking for a while about a specific known school, and then want to add that the school is no longer there, you should probably use this word order. For example if somebody asked "Что ты знаешь о школе?", then "школы уже нет" is an appropriate answer.
"уже нет школы" puts more emphasis in the school itself: it is interesting that it is a school, that which there is no more. You are more likely to use this word order if this is the very first time in the conversation that anybody mentions a school at all. For example if somebody asked "Каких зданий уже нет?", "уже нет школы" is appropriate as a response.
No, that would instead be "Нет, школе ещё нет". I see that you are also doing Spanish: You can think of "ещё" as Spanish "todavía / aún", and "уже" as Spanish "ya". That way, the sentences match perfectly
"Нет, школы уже нет" --> "No, ya no hay escuela"
"Нет, школы ещё нет" --> "No, todavía no hay escuela"