There is no doubt that Islam is one religion that was revealed by Almighty Allah to the Seal of Prophets (peace be upon him). He suffered much pain and harm in order to deliver the message of Islam to the nation. He directed his nation to the right direction after his death when he ordered them to guard the teachings of the Quran and the Ahlulbait (peace be upon them). He guaranteed that if both were adhered to, they will never be misguided. Part of the nation of Islam abandoned the Ahlulbait (peace be upon them) and the differences between the nation of Islam became a reality. We, as followers of the Ahlulbait (peace be upon them) should act upon their teachings and uphold their directions to the right path, while dealing with the rest of the Muslims with good manners and etiquettes, and not raise the differences or cause hatred against them, since we all believe in Almighty Allah and His Messenger and the common religious obligations, like prayers, zakat, fasting, pilgrimage, etc., in order to preserve the general entity of Islam. The teachings of the Imams (peace be upon them) emphasize this in various ways.
Yes, the hair of a cat is regarded as Tahir unless it is known to have become Najis. But the prayer is invalid based on an obligatory precaution if the individual has on his body or clothes any part of an animal whose meat is impermissible to eat - like a cat - even if it is hair or fur.