I think that a new portion to the FAQ should be added regarding authenticity. and what I mean is the authenticity of Ahadith, Shia it is or sunni, or any other sect, the authenticity of a Hadith should be strong. I think this portion should also mention that when quoting something from another sect, it should be made sure that it is authentic.
I agree that ahadith (etc.) that are clearly fabricated shouldn't be allowed, but I feel that limiting resources to only those which are strongly authentic unnecessarily cripples the site.
According to Imam Nawawi, there's ijma that a da'if hadith can still be referred to if good deeds are concerned (even if it is too weak to be used for fiqh). These hadith are still recorded in the hadith literature, and they still have their uses. Just because we can't be sure that it's authentic doesn't automatically mean that it's wrong.
I don't think authenticity is really an issue, rather the issue is more about credibility. If someone uses a weak hadith in a way that no credible scholar (from the relevant school) would have used it (e.g. in a matter of fiqh), it should be downvoted and possibly deleted. However, if they use it in a manner that a credible scholar would have (or has) used it, I don't see a problem.
Whether any particular use, or even scholar, is credible or not is highly subjective; to paraphrase Justice Potter Stewart, I could never succeed in intelligibly defining it, but I know it when I see it.
This is why voting is so important, and why attracting an expert audience is vital; they know credibility when they see it, and vote accordingly.
I don't think it should be a requirement.
It is too much work to verify every single hadith. Other readers should be encouraged to call out a weak or fabricated hadith in the comments or one translated from an unreliable site.
The weaker hadiths have their place and purpose, otherwise they would have been abandoned and forgotten. Many hadiths are quite valid even though they're not 'sahih', it just means that those hadith don't meet the strict standards that a 'sahih' one requires.
The purpose of a hadith also matters. A hadith used for historical/narrative purpose need not be as correct as one used as fiqh.
However as etiquette, the authenticity should be mentioned. At least the source of the hadith should be mentioned, as well as the source of translation. It's easier to reference and check. If one is not certain of the authenticity of a hadith, that should be mentioned in the answer as well.
I believe that inauthentic Ahadeeth, whether it is fabricated or da'eef it doesn't matter, it should not be allowed to be used. Inauthentic Ahadeeth were and are sources of deviation, and has caused many evils whether it is in matters of the unseen, or it in matters of shariah. those who use inauthentic Ahadith, it is feared that they will fall under the warning of the Prophet (peace and blessings be upon Him):
مَنْ كَذَبَ عَلَىَّ مُتَعَمِّدًا فَلْيَتَبَوَّأْ مَقْعَدَهُ مِنَ النَّارِ
Whoever lies upon me intentionally, then let him take his seat in the Fire
And even if they do not lie on the messenger directly, they still know that there is in-authenticity in a hadith. And this is pointed out by the saying of the prophet (Peace and blessings be upon Him):
كفى بالمرء كذبًا أن يحدث بكل ما سمع
It is enough for a man to prove himself a liar when he goes on narrating whatever he hears
And ibn Hibban said in his saheeh (Saheeh Ibn Hibban page 27):
Chapter/Fasl in mention of entering hell those who ascribe things to the messenger (peace and blessings be upon Him) while not knowing whether it is authentic or not.
And then he (authentic) mentions proofs to support.
See Silsilah al Ahadeeth Ada'eefah/ سلسلة الأحاديث الضعيفة و الموضوعة و أثرها السئ في الأمة By Sheikh Nasiruddeen Al Albany
Hence I believe it should be a requirement here, for those who quote Aadeeth in their answers, to make sure they are authentic. It really isn't that hard since many Ahadeeth have been checked regarding their authenticity already and there are books concerning what is authentic and what is not and why.