Based on what was the Earth's rotation around itself, around the sun, and around the galaxy considered a "scientific fact"? Is this considered mutawatir (transmitted by so many that it is beyond doubt) or among the necessities of reason? And what is the response to the religious evidences cited by العلامة Ibn Uthaymeen to support that the sun revolves around the Earth?

There is nothing in the Quran or the Sunnah that denies or affirms the Earth's movement and rotation. However, it has been reported through continuous transmission from Muslim astronomers and others that the Earth rotates and moves like other planets, and this is a definitively established fact.

The Earth is a planet that revolves around the sun in a year, and this revolution results in the four seasons. It also rotates around its own axis in 24 hours, and this rotation results in day and night. All of this occurs by precise divine decree, as stated in His Almighty's saying: "And all [celestial bodies] in an orbit are swimming."

The Earth's movement and rotation are neither rationally obligatory nor rationally impossible; rather, they are permissible. Specialists have proven them, and there is nothing in Islamic law that contradicts them. This issue does not entail any legal ruling or a prescriptive عمل (action), so research into it should be left to specialists.