What's the difference between "ag cur báistí" and "ag cur fearthainne"?
There isn't any difference. In Munster, you might come across ag báisteach or ag fearthainn (without the cur+genitive) and in Ulster and Connacht you might come across ag cur without any modifier.
Would 'yet' work here?
Yes — without context, fós could be translated as either “still” or “yet” in this sentence.
Unlikely fós implies the situation is the same;
since it is raining it is "still" raining whereas
nach bhfuil sé ag cur báistí fós? Isn't it raining yet
It is dry. Is it raining yet. It is wet. Is it still raining. They mean different things.
In every language, there are ambiguities. From this and other commentary, it seems like this is one of those situations for Irish.
ag cur báistí - literal translation 'putting rain'? (the literals, when they do come close, help me remember the construction)
Yes, essentially 'putting rain'.