What is the authenticity of the two aforementioned hadiths regarding the tribulations in Yemen, Sham, and Mesopotamia, and the statements attributed therein to the Prophet, may God’s prayers and peace be upon him?

The first Hadith, "When tribulations abound, cling to Yemen," is not authentic. There is a similar Hadith, "When tribulations abound, cling to the edges of Yemen," but its origin is unknown. However, it is narrated in the Sunan that the Prophet (peace be upon him) said: "Hold fast to Sham (Syria), for it is the choicest of Allah's lands, to which He gathers the choicest of His servants. Whoever refuses, let him join its Yemen (right side) and drink from its springs, for Allah has guaranteed Sham and its people to me."

The second Hadith has not been found in available sources.