Is it permissible to seek intercession or refuge through Fatimah Al-Zahra, and is the supplication: "O Allah, bless Fatimah, her father, her husband, and her sons, and the secret enshrined within her, by the number of all that Your knowledge encompasses," or "By the right of Fatimah and her father, and the secret enshrined within her," considered as ascribing partners to Allah (shirk)?

It is prescribed for a Muslim to seek refuge in Allah alone, as this is what the Quran and Sunnah have commanded. The prophets and messengers sought refuge in Allah, not in created beings. As for seeking refuge in created beings, it is among the acts of the جاهلية (pre-Islamic ignorance) and may be considered shirk (polytheism) if the matter from which refuge is sought can only be accomplished by Allah, because seeking refuge is an act of worship, and diverting worship to other than Allah is shirk. The questioner is advised to avoid people of innovation and stay away from their gatherings.