Ali ibn Abi-Talib did not get angry when Abi Bakr As-Siddiq was chosen Caliph instead of him because Ali greatly respected Abi Bakr, acknowledged his status with Prophet Muhammad (PBUH) and accepted people's choice.
A clear proof of Omar ibn Al-Khattab's sincere love for Fatimah Az-Zahraa and the Prophet's Family is evident in a Saying narrated by Aslam Al-Adw, who said:
"When Abi Bakr was recognized as Caliph, Ali and Az-Zubair ibn Al-Awwam went to Fatimah to consult with her. Omar heard about that, so he went to Fatimah and said, 'O daughter of the Messenger of Allah, we love your father more than any other person, and after him, we love you more than any other person alive. Then he talked to her.'
Then Ali and Az-Zubair came to Fatimah who said, 'Leave informed.' They left and only returned after they had pledged allegiance."
This is what really happened as narrated by one of the Companions and also conforms with the spirit and character of Omar ibn Al-Khattab, who dearly love the Prophet and his family.
Safavid Shiites twisted the events and made up narrations full of lies about Omar ibn Al-Khattab. They falsely claimed that Omar ibn Al-Khattab said to Fatimah bint Muhammad, peace be upon him, "If those two meet at your house (meaning Ali and Az-Zubair ibn Al-Awwam), I will burn up the house because they dissent from the rest of Muslims, having delayed paying allegiance so long." They also claim that when Omar left, Ali and Az-Zubair went to Fatimah who said to them, "Omar came to me and swore to burn down the house if you come here. By God he will do what he swore to do, so leave and never come back." So, they left and never came back until they had pledged allegiance.
Some even claimed that Omar beat Fatimah who was pregnant and her fetus, Mohsen, got aborted. This is wrong because Mohsen ibn Ali ibn Abi Talib died during the life of Prophet Muhammad. All the narrations of the Safavid Shiites in this concern are mere lies. Such harsh thing could not have been uttered bu Omar who dearly love Prophet Muhammad and all his family, not to mention his daughter Fatimah. Moreover, this narration represnt Ali ibn Abi-Talib as a coward who would tolerate such insults, although Ali was one of the most courageous Companions of Prophet Muhammad.
Safavid Iranians really hate Omar ibn Al-Khattab because he defeated their Emperor Yazdgird III and the Persian empire came to end at the hands of Muslim armies during Omar's Caliphate.
The Muslim army commander captured Yazdgird's daughter, but Omar refused to turn her to a slave girl and upon her conversion to Islam, she got marries to Al-Husain ibn Ali and she gave birth to his son, Ali Zain Al-Abedeen, who survived the Karbala massacre. Turkish writer Fawzi Tatakar mentioned that Omar married off the second daughter of Yazdgird to Mohammed bin Abi Bakr As-Siddiq and the third to Abdullah bin Othman ibn Affan.
This brings me to an important question: Would Ali have accepted to marry off his own daughter Omm Kulthum to Omar ibn Al-Khattab, had there been such enmity between them as claimed by the Safavid Iranians?!
Omar marrying the daughter of Ali is a clear evidence of the mutual love, appreciation and loyalty the two great Companions of the Prophet harbored for each other. This is a clear evidence on Omar's love for the family of the Prophet, just as he loved the Prophet and helped him wholeheartedly in his struggle to spread Islam.
Omar said to Ali while asking his daughter's hand for marriage, "By Allah, no other man would be a better company for her than me." Ali said, "True."
Omm Kulthum gave birth to a girl, Ruqayyah, and a boy, Zaid, who died young while attempting to reconcile between the sons of Adey ibn Kaab who were fighting one night. One of them hit him and fractured his skull. Zayd ibn Omar died on the spot and his mother Omm Kulthum was very sad. She died soon after him and was buried next to him.