Is it permissible for a girl to supervise an Islamic section in a mixed forum after performing Istikhara three times and feeling at ease, given her mother's refusal out of fear of her being labeled a "terrorist," and is the Istikhara she performed valid?

Among the best ways for a servant to draw near to Allah is by serving His religion and inviting others to it with wisdom. As a reward for what you exert in serving your religion, seeking Allah's pleasure, you should proceed on your path, as long as you have sought Allah's guidance (istikhara) and trustworthy individuals have advised you. You will find nothing but good. This must be done with your mother's approval and conviction, as obeying them is obligatory in what is permissible. Do not pay attention to the illusions and intimidations cast by enemies, and rely upon Allah. Remember His saying: "It is only Shaytan who frightens [you with] his allies; so fear them not, but fear Me, if you are [truly] believers." (Quran 3:175). And know that whoever exerts effort in serving the religion, Allah guides and helps him, as He says: "As for those who strive hard in Our cause, We will surely guide them to Our paths. And indeed, Allah is with the doers of good." (Quran 29:69). The important thing is to convince your mother of these truths.