Is it permissible for men and women to intermingle on Facebook for the purpose of spreading Islam? Is it permissible to imitate the disbelievers in this process? And is it permissible to use tangible, material evidence to prove the truth of Islam?

It is not permissible for men and women to mix on Facebook and similar platforms, nor is it permissible to imitate disbelievers in what is specific to them, even if it is claimed to be for the purpose of spreading Islam. There is no harm in using tangible material evidence to prove the truth of Islam; rather, it is beneficial. The important thing is the soundness of the evidence and the clarity of its indication. Indeed, Allah supported His prophets with signs and material miracles to establish the proof.