How sound are the narrations that speak of Aisha's shouting at the Prophet and its consequences, and of the Prophet's visit to Fatima and his helping her with grinding, and her conversation with Ali?

Firstly, it is essential for a Muslim to verify what they read and hear, as Allah Almighty commanded in His saying: "And do not pursue that of which you have no knowledge." This is especially true with the widespread circulation of fabricated hadiths on social media.

Secondly, regarding the first hadith, it is narrated in the books of Sunnah from Nu'man ibn Bashir that Abu Bakr sought permission to enter upon the Prophet (peace be upon him) and heard Aisha's voice raised. When he entered, he grabbed her to strike her, saying: "How dare you raise your voice at the Messenger of Allah (peace be upon him)!" The Prophet (peace be upon him) restrained him, and Abu Bakr left angrily. When Abu Bakr had departed, the Prophet (peace be upon him) said to Aisha: "How did you find me saving you from the man?" Abu Bakr stayed away for a few days, then sought permission to enter upon the Messenger of Allah (peace be upon him) again, and found them reconciled. He said to them: "Let me be included in your peace, just as you included me in your conflict." The Prophet (peace be upon him) said: "We have done so, we have done so." This hadith has been authenticated by many hadith scholars, such as Al-Albani. Its implication lies in the Prophet's (peace be upon him) guidance in dealing with anger by calming the angry person and not reciprocating their anger. Scholars have inferred that among the rights of brotherhood is to bear and absorb anger. In summary, the hadith is not cited to commend Aisha's blameworthy anger, but rather to illustrate the Prophet's guidance in dealing with blameworthy character.

Thirdly, concerning the second hadith, its original story in "Al-Sahihayn" is that Fatimah complained about the hardship she endured from the grinding mill. She went to the Prophet (peace be upon him) asking for a servant. He said to her and Ali (may Allah be pleased with both of them): "Shall I not tell you something better than what you have asked for? When you go to bed, say 'Allahu Akbar' thirty-four times, 'Subhan Allah' thirty-three times, and 'Alhamdulillah' thirty-three times. That is better for you than a servant." As for the wording mentioned in the question: "You are closer to my heart, and Ali is in my heart," it has no basis in the books of Sunnah and Hadith. It is a lie and a fabrication, and it is not permissible to circulate such fabricated hadiths except for the purpose of refuting and disproving them.