Sahih: Jewish boy who got sick accepted Islam.

Classification: Sahih (authentic)

From Anas (may Allah be pleased with him) who said: “A Jewish boy used to serve the Prophet (peace and blessings of Allah be upon him), and he fell sick. The Prophet (peace and blessings of Allah be upon him) went and visited him. He sat by his head and said, ‘Enter Islam.’ The boy looked at his father, who was present. (His father) said, ‘Obey Abu’l-Qaasim (meaning the Prophet peace and blessings of Allah be upon him).’ So the boy entered Islam and the Prophet (peace and blessings of Allah be upon him) left saying, ‘Praise be to Allah Who has saved him from the Fire.’”
Narrated by Al-Bukhaari (1356)

Fabricated: The neighbor who used to throw trash at the Prophet's door.

Classification: Fabricated.

The story goes something like this: The Prophet (Peace & Blessings of Allah be upon him) had a jewish neighbor who used to throw trash at his door every day. The Prophet (Peace & Blessings of Allah be upon him) was patient and did not do anything. One day, the Prophet noticed that there was no trash at his door, so he went to check on his neighbor who turned out to be sick. She asked him how did he know and he said because there was no trash at his door that day, so the woman was so touched that she accepted Islam.