In sum, the isnād is hasan. Allah knows best.
Under the heading “The virtue of the month of Ramadan and the excellence of good deeds therein,” he writes:
1 – On the authority of Abū Hurayrah: when Ramadan arrived the Prophet ﷺ said, “A blessed month has come to you … In it there is a night better than a thousand months; whoever is deprived of its good has indeed been deprived.” Aḥmad, al-Nasāʾī, and al-Bayhaqī narrated it.
I say: The chain is broken, for they all transmit it through Abū Qilābah from Abū Hurayrah, and, as al-Munḏhirī remarked, “To the best of my knowledge he did not hear from him.” Even so, the report is ṣaḥīḥ by virtue of corroborating evidence, for the clause concerning the gates of Paradise being opened, the gates of Hellfire shut, and the devils chained is established in the two Ṣaḥīḥs on the authority of Abū Hurayrah; it is also cited in al-Ṣaḥīḥah, no. 1307. The remainder of the wording appears in Ibn Mājah from Anas with a hasan isnād, a grading al-Munḏhirī affirmed.
4 – On the authority of Abū Saʿīd, the Prophet ﷺ said, “Whoever fasts Ramadan, knows its limits, and guards himself from what ought to be avoided will have his previous sins expiated.” Aḥmad and al-Bayhaqī narrated it “with a good chain.”
He says as much, but upon examination there is a problem: both isnāds contain ʿAbd Allāh ibn Qurayṭ. Al-Haythamī notes, “Ibn Abī Ḥātim mentioned him, yet recorded no criticism or endorsement.” Thus he is effectively unknown. Ibn Ḥibbān’s declaration of reliability does not dispel that obscurity, for— as we have clarified many times—his methodology is to grade an unknown transmitter trustworthy. Hence al-Ḥusaynī said in Rijāl al-Musnad, “Unknown,” and the same appears in al-Taʿjīl. The hadith is listed in al-Ḍaʿīfah, no. 5083.
5 – On the authority of Abū Hurayrah, the Messenger of Allah ﷺ said, “Whoever fasts Ramadan out of faith and seeking reward will have his past sins forgiven.” Aḥmad and the compilers of the Sunan narrated it.
The author has taken an unnecessarily long route, for the hadith is found in both Ṣaḥīḥs with this very wording, and it is cited in Irwāʾ al-Ghalīl, no. 906.