When quoting a book in a question that was originally written in a language other than English, is it preferable to quote it in the original version or in an English translation, or both? If one should supply a translation, should it be an official one of some kind or is it okay to do the translation oneself?
My strong recommendation is to always, if possible, include both.
If you have access to it, include original text.
Always include English translation, if nothing else, from Google Translate.
However, no matter what your translation source (professional published translation, your own translation as fluent speaker, or Google copy/paste), please cite where the translation is from.
Original source is needed because StackExchange is geared towards experts.
When working with a text, no translation is ever 100% precise (for a variety of reasons which exceed the margins of this answer). As such, a true expert analysis which relies on the text may require relying on the un-changed original, both for the person answering, the person who asked for an answer, and for later readers.
On some level this is similar to Skeptics.SE. A citation should be to primary research source, not to Wikipedia summary or to an article translating the research into popular wording.
The site is geared towards English speakers, like a vast majority of SE sites.
As such, most readers will NOT be fluent in the source language, so English translation is non-negotiable.
I'm fluently bilingual, and one of the few experts on Russian/Soviet works on Scifi.SE, with a reasonable body of work to show for it, which largely follows the above convention and was very well received by site users, even in obscure Russian works tags.
Here are examples of answers:
Add an English translation, at the least.
I've read some works by Dostoevsky, a Russian writer in the 19th century. Now, I'm absolutely not fluent in Russian - actually, you could count the number of Russian words I know on one hand, and still have four fingers and a thumb available. However, I was able to answer a question about Dostoevsky earlier today, including an English translation of part of one of his stories.
Stack Exchange is meant - with the exception of certain sites - to be used in English. I think we should apply this to quotes, as many people will be like me - having read translated versions, but knowing nothing about the original text. Otherwise, the quotes will be useless to a large fraction of visitors.
Ultimately, this seems like less a matter of site policy and more a matter of how to write questions to get the best answers. Both English and non-English sources are appropriate; you're just going to have more reliable success with English ones.
If the non-English version is particularly necessary, or the translation itself is the topic of conversation, or for any other reason the English version is unavailable, I'd say go ahead and use the non-English version. If you can, and you're confident enough that your translation won't hamper answerers, it's a good idea to include that, too.
But if you can, and it doesn't hamper your question in any way, try to stick to English. Literature.SE doesn't need to be an English literature only site, but SE as a whole is an English site.
This site is in English, so your post should be comprehensible by an English speaker.
If you quote a passage, indicate who the translator is (give both the name and the publication, if you have this information). If the translation is your own, say so. Translations can be misleading or wrong sometimes, so it's important to know where they're coming from.
I think it's generally not useful to include the original text, unless there's some doubt regarding the correctness of the translation.