Is the Hadith that mentions the Prophet's — may Allah's prayers and peace be upon him — weeping when passing by a graveyard and supplicating for the dead, and his saying that fasting one day and performing night prayers for one night in Rajab prevents the torment of the grave, authentic?

This hadith is not found in the books of Hadith. The author of the book "Nuzhat al-Majalis wa Muntakhab al-Nafais" mentioned it without a chain of narration (isnad), which indicates that it is fabricated and forged. Some scholars have ruled that hadiths about the specific فضيلة (virtue) of fasting Rajab are fabricated. Ibn Taymiyyah said: "As for fasting Rajab specifically, all its hadiths are weak, rather fabricated, and scholars do not rely on any of them. They are not among the weak hadiths that can be narrated for virtues; rather, most of them are fabricated lies."