Disclosure: I'm not an authority on Judaism nor am I Jewish. Therefore, my answer is necessarily incomplete. I'm also an active participant on Biblical Hermeneutics and a moderator pro tempore there. Our FAQ actively solicits answers from a Jewish perspective. Also, I consider myself your friend. Therefore, my answer is hopelessly biased and "too localized". If you think it won't help other people, feel free to delete this answer.
Perhaps the best way to tackle the question is by analogy. My local newspaper has a Religion section. One of the ways the paper fills that space is by running an "In Theory" column which invites local religious leaders (including atheists) to answer questions prompted by current events. For instance, when the Federal budget was making headlines last summer, the paper asked: Does the budget need a dose of morals? The various pastors, rabbis, atheists, and imams decided whether or not to respond. Those that did wrote up answers for publication.
One rabbi who participates frequently in this column is Rabbi Simcha Backman of Chabad of Glendale, who also happens to be the director of AskMoses.com.
So I think another way to ask your question is:
Under what conditions may a Jew answer questions about the Bible in their local newspaper?
The parallel would be:
Stack Exchange Incorperated => a local newspaper
Biblical Hermeneutics => a column that asks Bible questions of clergy
Even Christianity.SE has a formal policy of being a secular site, so Biblical Hermeneutics is even more of one. Stack Exchange is in the business of selling advertising and job listing in order to make a profit. Our sites, including Mi Yodeya, are a means to that end. We provide content that SEI uses to sell its product. This business model is so similar to a newspaper it's eerie.
I don't think the contributors to the "In Theory" column are paid. Rather, they contribute in order to see their name in print, sway public opinion, or to advertise their organization. We aren't paid either, but participate in order to get an endorphin rush when our reputation number goes up, learn something about a topic we are interested in, or because we meet like-minded people. The motivations seem about the same on both sides of the equation.
Since rabbis, such as Rabbi Backman, do participate in the column about current events, I would expect that it would be fine to answer questions about the Tanakh. Sometimes the questions are about "family matters" specific to Christianity. (Such as: Have Christians retreated from the world?) Even in these cases, at least one local rabbi regularly weighs in on such topics. So I have to assume that he sees no problem with what I think it an analogous situation.
There is, however, a difference between the column in the newspaper and Biblical Hermeneutics.SE. In the paper (and the web version) each answer is clearly marked with a name, religious tradition, and organization. (The atheists tend to just use their city for the last item.) On Stack Exchange, we get a user name, reputation, and badge counts. If you meet certain criteria, your user card will be expanded if a reader happens to
leave the pointer in the wrong place hover over your avatar. Unless you indicate your religious tradition in the answer or your profile's About Me field, there's no way to tell who you are. Further, there's no way to force other people to disclose their affiliation. Therefore the relationship between various answers and religious doctrine must be deciphered by the reader. Other interactions on the site (commenting, editing, chatting, voting to close, etc.), can also be misinterpreted in various ways.