The vastness of Allah's mercy and encouragement to show mercy

It is narrated from Sayyiduna Abdullah bin Abi Awfa al-Aslami (may Allah be pleased with him) that he said: I came to the Messenger of Allah (peace and blessings be upon him), and Sayyiduna Abu Bakr and Sayyiduna Umar (may Allah be pleased with them both) were sitting with him (peace and blessings be upon him). Suddenly a small child was seen crying, so the Prophet (peace and blessings be upon him) said to Sayyiduna Umar (may Allah be pleased with him): "Hold this child close to you, for he has lost his way." So Sayyiduna Umar (may Allah be pleased with him) held him to his chest. We were still sitting when a woman appeared, wailing; I think she was calling out, "Wa bunayyah!" (Alas, my child!) and weeping. The Messenger of Allah (peace and blessings be upon him) said: "Call this woman, for she is the mother of this child." The woman was so distraught and restless in grief for her son that, from the intensity of her grief, her head was uncovered and there was no head-covering on it. She came near and, weeping, took the child from the lap of Sayyiduna Umar (may Allah be pleased with him). The child was still in her lap when she turned, and as soon as her eyes fell on the Messenger of Allah (peace and blessings be upon him), she cried out (in shame and modesty): "Wa khizyah! (Alas, my disgrace!) Am I seeing the Messenger of Allah (peace and blessings be upon him) (in this state)?" At this the Messenger of Allah (peace and blessings be upon him) asked: "Do you find this woman extremely kind and tender towards her child?" The Companions (may Allah be pleased with them) said: "Yes, O Messenger of Allah (peace and blessings be upon him)! This state of hers is enough as a witness to her motherly love and mercy." Then he (peace and blessings be upon him) said: "By the One in whose Hand is my soul! Allah Almighty is certainly far more merciful to the believers than this woman is to her child."
[Musnad Abd bin Humayd: 524; al-Mu'jam al-Kabir of al-Tabarani: 8133; al-Hafiz al-Busiri said: Its chain is hasan (good).]

It is narrated from one of the Companions (may Allah be pleased with him) of the Prophet (peace and blessings be upon him) that he said: We were with the Prophet (peace and blessings be upon him) on a journey, and some of the people caught the chick of a sparrow. At this the sparrow (restless with love) began to fall upon their saddles, so the Messenger of Allah (peace and blessings be upon him) ordered that its chick be returned to it. Then he (peace and blessings be upon him) said: "Allah is certainly more merciful to His servant than this sparrow is to its chick."
Al-Harith bin Abi Usamah narrated it in his Musnad (Bughyat al-Bahith: 1045). Its basis is narrated in Sahih Muslim (2754) from Sayyiduna Umar (may Allah be pleased with him) with the incident of a captive woman, while this incident of the sparrow is also found in Sunan Abi Dawud (2675) and Musnad Ahmad (4400).

It is narrated from Sayyiduna Anas bin Malik (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) said: "By the One in whose Hand of power is my soul! Allah Almighty sends down His mercy only upon the one who is merciful." The Companions (may Allah be pleased with them) said: "O Messenger of Allah, every one of us shows mercy." He (peace and blessings be upon him) said: "What is meant is not that particular mercy of yours (which you show to your own circle or your own people), until you become merciful to all people in general."
Takhrij: Musnad Abi Ya'la: 2827. Its chain contains Muhammad bin Ishaq, who is saduq (truthful) and a mudallis; however, on the basis of other supporting narrations this report is of the hasan (good) grade.

It is narrated from Abu al-Azhar that a man passed by the Prophet (peace and blessings be upon him) with two chicks of a bird while their father (the male bird) was circling over them. The Messenger of Allah (peace and blessings be upon him) said (at his callousness): "You have taken these two chicks while their father is circling around them. Why did you not leave one of them for it, so that 'its eye would be cooled by it' (and it would find peace of heart)?"
Al-Harith bin Abi Usamah narrated it in his Musnad. Its chain contains the narrator Ibn Lahi'ah, who is da'if (weak) due to the weakness of his memory; however, its meaning is supported by other sahih hadith concerning showing mercy to birds and not harming them.

It is narrated from Sayyiduna Sa'id al-Ansari (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) said: "Indeed, Allah, the Blessed and Exalted, will show special mercy to His believing servant on the Day of Resurrection because of the mercy and compassion he showed to a sparrow."
Al-Bayhaqi narrated it in Shu'ab al-Iman; however, its chain contains the narrator Abd al-Ghafur bin Abd al-Aziz, who is severely da'if (weak) and matruk (abandoned), because of which this report is technically weak.