Is it authentic that the Prophet (peace and blessings be upon him) prayed for 'Abd Allah ibn Abi, and regarding 'Umar's stance that he wanted to prevent him from doing so, and that the Qur'an was revealed in support of 'Umar?

The Shaykh : As for the prayer of the Prophet (peace and blessings be upon him) for this hypocrite, it was based on his independent legal reasoning (ijtihad). The Messenger (peace and blessings be upon him and his family) used to apply the rulings of the Shari'ah based on outward appearances. This man prayed and participated with the Muslims in many acts of obedience and worship, but as Allah the Exalted said: , so whatever the case may be, as the poet also said:
"And whatever trait a man may possess, *** Even if he thinks it is hidden from people, it will be made known."
So, however much these people conceal their hypocrisy, some words will inevitably slip from the tips of their tongues that indicate the disbelief that has settled in their hearts. There is no doubt that the disbelief of these people was doctrinal, a disbelief that makes one of them subject to the famous verse: . So this man was among them.
However, the Messenger (may Allah's prayers and peace be upon him), although he knew some of them, treated them as Muslims in what was apparent. This is part of the rulings of the Shari'ah. From this, the scholars derived the famous principle among jurists, which some ignorant people mistakenly think is a hadith, namely their saying: "We judge by appearances, and Allah takes care of the secrets." This is a correct jurisprudential principle, but it is not a Prophetic phrase. Rather, it is derived from many verbal and practical hadiths on the one hand, and from the application of the Messenger (peace be upon him) of this principle in his approach to dealing with all people, including hypocrites like this one.
So the Messenger (peace and blessings be upon him) prayed for him based on this principle, knowing that he was a hypocrite. The Prophet (peace and blessings be upon him) had one man among the Companions whom he chose and singled out with a special status above all his Companions, and who was the keeper of his secret, namely Hudhayfah ibn al-Yaman (may Allah be pleased with him). So he knew the names of the hypocrites. Therefore, 'Umar ibn al-Khattab, out of his fear of his Lord, would ask Hudhayfah, invoking his Lord, whether his name had been mentioned among the group of hypocrites whom the Messenger (peace be upon him) had named for him, and he would reassure him that he was not among them. In summary: the prayer of the Messenger (peace be upon him) for this hypocrite is a branch of his treatment of him as a Muslim, based on the aforementioned principle: "We judge by appearances, and Allah takes care of the secrets."