As of today, we don't have much of a formal policy on all these issues. Perhaps this would be the right time to come up with something. This long monologue is my take on all your questions...
Can I write בעברית?
The way I see it, there are two potential issues with writing in Hebrew:
Will the user's browser, operating system, device support it?
I doubt we should worry about "plain" hebrew. Lack of Hebrew support on modern computers is virtually non-existent. Even most modern-day smartphones have no trouble rendering simple Hebrew.
The problem really shows up when it comes to nekudos which may be more problematic. I would recommend steering clear of these for the time being, unless absolutely necessary.
Will the user be able to read it?
Not everyone can read Hebrew. By writing things in hebrew, we tend to place an additional barrier to understanding for such people. In other words, it's hard enough to understand the words, and by putting something in Hebrew we may be pushing Hebrew-illiterates away from our site.
On the other hand, there is one very big advantage to writing in Hebrew:
There is only one way to spell a word
When you type something in Hebrew, spelling is perfectly consistent. There are not two way to spell the word ברכה. But once you hit the transliteration step, confusion abounds. Is it beracha, berachah, bracha, brocha, brocho, berakhah, or brachah? I honestly don't know.
To get around this problem, I would sometimes purposely spell the same word differently throughout my questions, which would increase the possibility of getting a hit when searching. But this is quite a pain, and not very practical.
Ideas are still buzzing about in my head. Please wait for them to settle, and I will edit this post accordingly.......
How about quotes?
Quotes, in my opinion should always be in their original form (that's why it's called a quote). So if you're quoting the Torah, put it in Hebrew (again, I would go easy on the Nekudos unless necessary).
(note: I refer here to quotes, and not paraphrases)
Should I translate all non-obvious Hebrew words, even those transliterated into English, like berakhah?
This question really ties into the work on the glossary which is still being formulated.
In short, if the word is a tag, or can potentially be a tag, don't translate it. If not, find a place to link it to that would explain it. But it really depends, so your use own discretion if it's worthy of an inline translation...
Also see: Issac's blog post about jargon guidelines and the related meta discussion on Jargon use...
How about terms which have popular English translations, such as Bereishit vs Genesis?
Yes, definitely put them in.
Are there transliteration guidelines? [...]
I think I covered this above, but comment if not...