Is the circulating story true that Lady Fatimah complained to the Prophet, peace and blessings be upon him, about Ali abandoning her bed, and that the water cooler spoke, saying she had bathed from it forty times, and the Prophet then told her, "The coolness, O Fatimah"?

The story mentioned in the question has not been found in any book; rather, it is a fabricated and forged tale stemming from a corrupt imagination that depicts the Prophet, peace and blessings be upon him, in a negative light. The correct account, as narrated in the books of Sunnah, is that a disagreement occurred between Ali and Fatimah, may Allah be pleased with them, and the Prophet, peace and blessings be upon him, reconciled between them, as narrated in the hadith of Sahl ibn Sa’d, may Allah be pleased with him: (The Messenger of Allah, peace and blessings be upon him, came to Fatimah’s house and did not find Ali there. He asked, 'Where is your cousin?' She replied, 'Something happened between him and me, so he became angry with me and went out, and he did not nap at my house.' The Messenger of Allah, peace and blessings be upon him, said to a person, 'Look for him, where is he?' He came and said, 'O Messenger of Allah! He is sleeping in the mosque.' So the Messenger of Allah, peace and blessings be upon him, came while he was lying down, and his cloak had fallen off his side, and he had dust on him. The Messenger of Allah, peace and blessings be upon him, began to wipe it off him, saying, 'Get up, Abu Turab, get up, Abu Turab'). It is not permissible to narrate fabricated stories, as a Muslim is accountable for what he transmits, due to the Almighty’s saying: (And do not pursue that of which you have no knowledge. Indeed, the hearing, the sight, and the heart - about all those [one] will be questioned) Al-Isra/36.