Can it be understood from the fact that the Prophet (peace and blessings of Allah be upon him) got angry on behalf of Faatimah that she was infallible?
Question

One of the Raafidis presented me with a specious argument, which is the view that Faatimah (may Allah be pleased with her) was infallible, based on the words of the Prophet (peace and blessings of Allah be upon him): “Faatimah is a part of me and whoever angers her angers me.” He said that the anger of Faatimah (may Allah be pleased with her) is part of the anger of the Messenger of Allah (peace and blessings of Allah be upon him), and the anger of the Messenger of Allah (peace and blessings of Allah be upon him) is part of the anger of Allah, hence Faatimah could not get angry for any idle reason, i.e., she was infallible. How can we answer this specious argument?.
Praise be to Allah.The Raafidis are a people who like to argue and stir up specious arguments which they exaggerate and present them as facts to the common folk in order to prove their point. But the one whom Allah has blessed with understanding of His religion and the ability to ask scholars, will undoubtedly realize the falseness of their arguments and the insignificance of their evidence. That includes the specious argument mentioned by the questioner. This argument may be answered in several ways: 
1 – The Raafidis quote this hadith as evidence, as is mentioned by the questioner, to reach the conclusion that Abu Bakr (may Allah be pleased with him) made Faatimah angry when he did not give her what she asked for of the inheritance of the Prophet (peace and blessings of Allah be upon him). If that is what happened, and Abu Bakr made Faatimah angry, then he also made the Prophet (peace and blessings of Allah be upon him) angry, and thus made Allah angry too! This is a sign of their ignorance and folly, for the hadith was originally narrated concerning ‘Ali (may Allah be pleased with him). Al-Bukhaari and Muslim narrated that al-Miswar ibn Makhramah said: ‘Ali proposed to the daughter of Abu Jahl. When Faatimah heard of that she came to the Prophet (peace and blessings of Allah be upon him) and said to him: Your people are saying that you do not get angry for the sake of your daughters, and ‘Ali is going to marry the daughter of Abu Jahl. The Messenger of Allah (peace and blessings of Allah be upon him) stood up and I heard him when he recited the shahaadah, then he said: “I gave a daughter of mine in marriage to Abu’l-‘Aas ibn al-Rabee’, and when he spoke he told me the truth. Faatimah bint Muhammad is a part of me, and I do not like for her to be hurt. By Allah, the daughter of the Messenger of Allah and the daughter of the enemy of Allah will not be joined together as wives of one man.” So ‘Ali abandoned that proposal. According to a version narrated by al-Bukhaari: “Faatimah is part of me and whoever angers her angers me.” [al-Bukhaari no. 3523 and 3556; Muslim, no. 2449]. 
By examining the report quoted above it will be clear that if blame were to be laid on anyone, it would be laid on ‘Ali (may Allah be pleased with him), because the reason why this hadith was narrated was his wish to propose marriage to the daughter of Abu Jahl, as the result of which Faatimah (may Allah be pleased with her) became angry. It is established among the scholars of Usool that what counts is the general meaning of the words, not the specific reason for them, but it is also established that the reason for the text must be taken into account first and foremost and the reason cannot be excluded at all. When the Raafidis quote this hadith as evidence for criticizing Abu Bakr (may Allah be pleased with him), because of their ignorance and trickery, they conceal the fact that criticism – if there is any criticism – would be directed at ‘Ali first of all. 
Secondly: The anger mentioned in the hadith occurred for a specific reason which is mentioned above. This indicates that the anger arose not because Faatimah was infallible or whatever claims the Raafidis make, but because the Prophet (peace and blessings of Allah be upon him) was keen to protect his daughter’s feelings, and what upset her also upset him (peace and blessings of Allah be upon him). This is indicated by what is mentioned in the report of Muslim: “Faatimah is a part of me and what hurts her hurts me.” This was offence caused to the Prophet (peace and blessings of Allah be upon him) and it has nothing to do with the matter of infallibility as those Raafidis claim. 
Thirdly: The Prophet (peace and blessings of Allah be upon him) said in other saheeh ahaadeeth: “Whoever obeys me had obeyed Allah and whoever disobeys me has disobeyed Allah. Whoever obeys my commander has obeyed me and whoever disobeys my commander has disobeyed me.” Narrated by al-Bukhaari, no. 6718; Muslim, no. 1835. According to consensus – even among the Raafidis – this does not mean that the commander is infallible; rather some of the commanders sent by the Prophet (peace and blessings of Allah be upon him) made mistakes in some matters which it is known went against the Book of Allah and the Sunnah of His Prophet (peace and blessings of Allah be upon him). An example of that is the proven report in which ‘Ali (may Allah be pleased with him) said: The Prophet (peace and blessings of Allah be upon him) sent out an expedition and he appointed in charge of them a man from among the Ansaar and told them to obey him. He got angry and said: Didn’t the Prophet (peace and blessings of Allah be upon him) tell you to obey me? They said: Yes. He said: Then gather firewood for me. So they gathered it for him and he said: Light a fire. So they lit a fire. He said: Enter it. So they thought of entering it, but some of them held others back and said: We have fled to the Prophet (peace and blessings of Allah be upon him) to escape the fire. They stayed until the fire burned out and his anger died down. News of that reached the Prophet (peace and blessings of Allah be upon him) and he said: “If you had entered it you would never have come out of it until the Day of Resurrection. Obedience is only with regard to that which is right and proper.” Narrated by al-Bukhaari, no. 4085; Muslim, no. 1840. Hence the Prophet (peace and blessings of Allah be upon him) restricted this obedience only to that which is right and proper. So if the anger of Faatimah was part of the anger of the Messenger (peace and blessings of Allah be upon him), then according to consensus it is limited only to that which is right and proper, and if the anger of Faatimah was for reasons that were not in accordance with the law of Allah, then what is right and proper is to implement the law of Allah, even if it angered Faatimah. Hence the Prophet (peace and blessings of Allah be upon him) said: “If Faatimah the daughter of Muhammad were to steal, I would cut off her hand.” And he (peace and blessings of Allah be upon him) said: “O Faatimah bint Muhammad, save yourselves, for I cannot avail you anything before Allah, ask for whatever you want of my wealth.” This implies that she was not infallible. 
See Manhaaj al-Sunnah al-Nabawiyyah, 4/250.