Is it permissible to communicate with men via forums and religious websites within the bounds of necessity for exchanging scholarly benefits, and is it permissible to post instructive topics in sections designated for women on the same forums?

The fundamental principle concerning a woman's participation in online forums and her conversations with men over the internet is permissibility, as long as it serves a beneficial purpose and adheres to Sharia-compliant guidelines. However, when a mother tells her daughter, "Do not speak to or message anyone you don't know," obeying her becomes obligatory. Disobeying this command is not permissible, especially since it closes doors to temptation. The examples you cited for necessity do not justify disobeying the mother's command. Therefore, she should restrict her conversations to those she knows.