Was Omar ibn al-Khattab the one who prohibited Nikah al-Mut'ah (temporary marriage)? And are the Ahl al-Bayt infallible, based on the noble verse: "Allah only intends to remove from you the impurity [of sin], O people of the household, and to purify you with [complete] purification"?

The Prophet, peace and blessings be upon him, is the one who prohibited temporary marriage (nikah al-mut'ah), not Umar ibn al-Khattab. As for those who argue for the infallibility of the Ahl al-Bayt (People of the Household) by quoting Allah's saying: "Indeed, Allah intends to remove from you all uncleanness, O people of the household, and to purify you with a thorough purification" (33:33), Shaykh al-Islam Ibn Taymiyyah responded that Allah's intention in this verse, like other verses that include Allah's love and pleasure with what is willed, does not imply that He has already created or decreed this willed outcome. The proof for this is that the Prophet, peace and blessings be upon him, after the revelation of this verse, supplicated for his household to have uncleanness removed from them and to be purified. If the verse were a declaration that uncleanness had already been removed from them and they had been purified, there would have been no need for his supplication. This is even clearer according to the view of the Qadariyyah. According to the view of Ahl al-Ithbat (those who affirm divine attributes and actions), there are two types of divine will in the Book of Allah: a legislative, religious will that includes love and pleasure, and a cosmic, predetermined will that includes creation and decree. This verse belongs to the first type, and what Allah wills (in this sense) may or may not occur, as in His saying: "And Allah wants to accept your repentance" (4:27), yet among them are those who repented and those who did not. Even if the verse indicated their purity, it does not imply infallibility from error.