Grading: This hadith is sahih; this chain is "hasan" in the context of corroborations and supporting reports, on account of Fulayh bin Sulayman and his two shaykhs.
Reference / Source: Imam Ahmad (41/ 25014) and Abu Dawud (3189) narrated it from Sa'id bin Mansur with this same chain, but in both of them the narrator's name "Muhammad bin 'Abbad" has been switched into "Muhammad bin 'Abd Allah bin 'Abbad" (an inversion of the name).

Reference / Source: Imam Ahmad (24498) and Ibn Majah (1518) narrated it via Yunus bin Muhammad, and Imam Ahmad (24499) via Surayj bin al-Nu'man, both from Fulayh bin Sulayman. However, Yunus did not mention Muhammad alongside Fulayh in his narration.

Reference / Source: Imam Muslim (973) (99) and (100), al-Tirmidhi (1033) and al-Nasa'i (2105 and 2106) narrated it with the chain of 'Abd al-Wahid bin Hamzah, from 'Abbad bin 'Abd Allah bin al-Zubayr. Imam al-Tirmidhi called it "hasan".

Reference / Source: Imam Ahmad (42/ 25357) narrated it via 'Abd al-Rahman bin 'Abd Allah bin al-Zubayr; Imam Ahmad (43/ 26245) and Ibn Hibban (3065) via Hamzah bin 'Abd Allah bin al-Zubayr; and Muslim (973) (101), Abu Dawud (3190), Ibn Hibban (3066), Abu Nu'aym in "al-Ma'rifah" (3306) and al-Bayhaqi (4/ 51) via Abu Salamah bin 'Abd al-Rahman; all three of them narrate from 'A'ishah (may Allah be pleased with her).
Details of the narration: In the narration of Abu Salamah the wording is: "The Messenger of Allah (peace and blessings be upon him) prayed the funeral prayer in the mosque over the two sons of Bayda': Suhayl and his brother."