Translation:Would you know where the nearest toilet is?
I think "don't you know" might be too literal a translation of the negative. The feel is more like "you wouldn't happen to know"
"Don't you (know)" in English is used for emphasis, and it often expresses at least a degree of surprise, frustration, or incredulity. If I accidentally walked into a broom closet when looking for the bathroom (restroom, toilet...) in my own home, someone might say to me, "Don't you know where the bathroom is in your own house?" (E.g., "Don't you know what day it is today? It's my birthday." "Don't you know how to do the simplest things?")
The "don't" / "doesn't" form can also be used for other questions in English to express some degree of doubt, in addition to the above (e.g., "Don't they live around here? I'm not sure."); and/or a sense that you had thought something was true ("Don't you have a cat? I thought I saw one in your living room last weekend.")
This isn't a comprehensive list; it just suggests some examples.
I have the sense that negative questions in Czech can simply have a more polite connotation than direct, positive questions. (E.g., as tylerskarz suggested, "You wouldn't happen to know...", or something similar.)
Jmm... Actually the point is that I was understanding the phrase as a doubt phrase. But, it seems that, for czech, it has a polite meaning. Anyway, it would be great to know if czech negative questions have also that "doubt" connotation.
Any czech could solve this doubt? :)
If this is a formal phrasing, isn't the nevite more appropriate than nevis?
No, you can ask anyone like this. It is completely fine this way if you are on the T side of the https://en.wikipedia.org/wiki/T%E2%80%93V_distinction with this person (tykání).
It seems weird to me that "Would you know" is the preferred translation to "Do you know".
As a native speaker I would never say just "Would you know", I would only say "Would you happen to know". Is this just me?
Okay, for that case maybe it is just me. But I do think the preferred translation should be "Do you know" https://books.google.com/ngrams/graph?content=Would+you+know+where%2CDo+you+know+where&year_start=1500&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2CWould%20you%20know%20where%3B%2Cc0%3B.t1%3B%2CDo%20you%20know%20where%3B%2Cc0