Is the Wailing Wall part of the Temple Mount?

Yes, it is the only remains of the wall surrounding the temple mount. According to ancient Jews history, temple mount was the site for the first and second temple which was destroyed in 587-596 BCE and 70ce. The wall is regarded as a very sacred place by the Jews. The wall now forms part of a larger wall that surrounds the Muslim Dome of the Rock and Al-Aqsa, and this has often resulted in disputes between Jews and Arabs.