To what extent is the hadith: "Indeed, the killing of Husayn (peace be upon him) has a warmth in the hearts of believers that never cools" authentic according to the Sunni school of thought?

This statement has no basis in the books of Hadith. The Rafidah (Shi'a) alone mentioned it, and what they alone mention is either weak or fabricated. The believer is distressed and saddened by the killing of Al-Hussein, may Allah be pleased with him, because he was killed unjustly, and he is the grandson of the Prophet, peace and blessings be upon him, and his killing incited discord. However, the believer's grief is not excessive; rather, he says "Innā lillāhi wa innā ilayhi rājiʿūn" (To Allah we belong and to Him we shall return) and remembers that Al-Hussein was martyred and promised Paradise. So, he does not slap his face or tear his garments.

Ibn Taymiyyah said that the killing of Al-Hussein, like the killing of Uthman, caused great discord. He was killed unjustly as a martyr, but an ignorant group takes Ashura as a day of mourning, sadness, and wailing, displaying the rituals of pre-Islamic ignorance such as slapping faces and tearing garments. The divine command regarding calamities is patience, seeking reward from Allah, and saying "Innā lillāhi wa innā ilayhi rājiʿūn," as Allah Almighty said: (And give good tidings to the patient, who, when afflicted with calamity, say: "Indeed we belong to Allah, and indeed to Him we shall return." Those are the ones upon whom are blessings from their Lord and mercy. And it is those who are the [rightly] guided). The Prophet, peace and blessings be upon him, forbade slapping cheeks, tearing garments, and calling out the cries of pre-Islamic ignorance. The believer who remembers an old calamity and says "Innā lillāhi wa innā ilayhi rājiʿūn" receives a great reward.