Is it permissible to create a mixed-gender da'wah (Islamic propagation) page on Facebook and interact therein with men and women, knowing that the pages of shaykhs and preachers are also mixed?

There is no harm in creating a page and making it available for both genders to like, as long as there is no forbidden communication. There is also no harm in responding to men's messages for a beneficial purpose, provided that one adheres to the শরীʿআ (Islamic law) guidelines, and that the response is limited to the extent of the need, with its phrases kept within the bounds of decorum, without inciting any discord.