Among the most complete of the believers in faith are those best in character and kindest to their families

It is narrated from Sayyidah Aishah (may Allah be pleased with her) that the Messenger of Allah (peace and blessings be upon him) said: "The most complete of the believers in faith is the one who is best of them in character and the most gentle and kind towards his family."
All the narrators of this hadith are trustworthy on the condition of the two Shaykhs, but they did not record it with this wording.

It is narrated from Sayyiduna Ibn Abbas (may Allah be pleased with them both) that Quraysh said to the Prophet (peace and blessings be upon him): Pray to your Lord to turn (the mount of) al-Safa into gold for us, and we will believe in you. He ((peace and blessings be upon him)) asked: "Will you do so?" They said: Yes. So he ((peace and blessings be upon him)) prayed, and Jibril (peace be upon him) came and said: "Allah sends you greetings of peace and says: If you wish, al-Safa will become gold, but whoever disbelieves after that, I will punish him with a punishment with which I have not punished anyone in all the worlds; and if you wish, I will open for them the gates of repentance and mercy." He ((peace and blessings be upon him)) said: "Rather, the gate of repentance and mercy (is better)."

It is narrated from Sayyiduna Ibn Abbas (may Allah be pleased with them both) that the Prophet (peace and blessings be upon him) preferred the gate of repentance and mercy over al-Safa turning into gold.
This is a sahih and preserved (mahfuz) hadith from the narration of al-Thawri from Salamah bin Kuhayl. Imran bin al-Hakam al-Sulami is one of the senior Tabi'in who is used as evidence. The two of them (Bukhari and Muslim) left this hadith out only - and Allah knows best - because of a discrepancy in its chain from Yahya bin Salamah bin Kuhayl, and Yahya makes many errors in narrating from his father.

It is narrated from Sayyiduna Ibn Abbas (may Allah be pleased with them both) that Quraysh said: O Muhammad, pray to your Lord to turn al-Safa into gold, and we will believe you. the Messenger of Allah (peace and blessings be upon him) asked: "Will you do so?" They said: Yes. Then Jibril (peace be upon him) came to the Prophet (peace and blessings be upon him) and said: "Take a firm pledge from them." Then Jibril (peace be upon him) said: "O Muhammad, Allah has granted you what you asked. If you wish, al-Safa will become gold for you, but whoever disbelieves after that, I will punish him with a punishment with which I have not punished anyone in all the worlds; and if you wish, I will open for them the gate of repentance and turning back." The Messenger of Allah (peace and blessings be upon him) said: "The gate of repentance and mercy is dearer to me."
This error (the change of name) does not weaken the hadith of al-Thawri, because I do not know anyone named Imran bin al-Ja'd among the Tabi'in; rather, Isma'il bin Abi Khalid narrated from Imran bin Abi al-Ja'd, while Imran bin al-Ja'd is from the generation after the Tabi'in (Atba' al-Tabi'in).

It is narrated from Sayyiduna Abu Musa (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) said: "Whoever does an evil deed and feels it to be bad while doing it, and does a good deed and is pleased by it, is a believer."
This hadith is sahih on the condition of the two Shaykhs, but they did not record it with this wording. I have already mentioned, in the sermon of Sayyiduna Umar (may Allah be pleased with him) at al-Jabiyah, that the two of them did not record it, and this (narration) is also in a different wording from that one.