I think that questions about Hebrew and Jewish languages should be allowed. Knowledge of Hebrew/JL is useful for studying Torah and Jewish texts, even though requiring questions about it to be asked specifically with regard to a specific passage in a Jewish text (as was expected on Mi Yodeya) can sometimes be burdensome. If this site were an ideal "community of experts," users of this site would be particularly well-positioned to answer these kinds of questions, since knowledge of Judaism and Hebrew/JL often come together.
I see no reason to limit them to a separate category. Hebrew/JL questions already overlap very much with the already diverse set of questions that could be considered Judaism. Restricting them to a category is arbitrary; if they aren't "Jewish enough," the same argument could be made for all kinds of other categories of questions, such as e.g. product recommendation (as opposed to Purim Torah, whose restriction to a category would serve a purpose: so as not to be confused with more serious questions). Hebrew/JL questions could simply be identified by a tag.
(On the other hand, A A's cryptic suggestion to allow posts in Hebrew might in fact warrant a separate category for questions written in Hebrew, whether about Judaism or Hebrew language. I don't know whether there is enough support for such an idea, though; it depends on whether there are enough Hebrew-speaking users who would participate.)
Is Modern Hebrew treated differently than Rabbinic, Mishnaic, Biblical Hebrew, etc.?
Are other languages of the Jews, such as Aramaic, Yiddish, and Ladino, treated similarly?
All of the above should be permitted, as Modern Hebrew and the others all have a Jewish literary tradition.
What about comparative Ancient Near Eastern linguistics?
Insofar as they relate to Hebrew or Jewish languages.
Are questions like "how does one conjugate this verb into this tense" on topic?
Are questions like "what does this Hebrew tattoo mean" on topic?
Are questions like "please translate this sentence into Hebrew" on topic?
Some of these might look like bad/uninteresting questions, but that would depend on how much work and research was put into the question. Bad questions are obviously not particularly wanted, but the scope itself may allow for them. Other policies (e.g. no translation requests for lengthy texts/without research) could potentially be implemented if inundation with this kind of question becomes a problem.