What is meant by the "People of the House" (Ahl al-Bayt), and does their virtue - if the hadiths mentioned in that regard are authentic - contradict the noble hadith: "An Arab has no superiority over a non-Arab except by piety"?

Whoever's lineage to the Household of the Prophet (Ahl al-Bayt) is established, has the right to respect, appreciation, and loyalty, as long as they are upon the religion of Islam. The 'Itrah (progeny/family) of the Prophet, peace and blessings be upon him, refers to the knowledgeable and righteous scholars. It is not a necessity that they be the best of people, because superiority in the sight of Allah Almighty is based on piety and good deeds. Indeed, superiority may belong to someone who is not from the Household over someone who is from them, if the former is more pious to Allah, like Abu Bakr Al-Siddiq, may Allah be pleased with him.