"Caucasian" is not a thing that exists. It comes from a disproven and thus obsolete classification-system by the late 18th century German scientists Meiners and Blumenbach. It's about as relevant to genetics of either ancient or modern-day humans as flat-earth theory is to modern space exploration and astronomy.
No. While there were probably "white" rulers of Egypt at times. Cleopatra was of Macedonian (greek) decent and thus most probably olive-skinned (just like the Macedonians that settled north of the mediteranean in contempary greece and balkan countries during these times).
King Tut was also not white. He was originally thought to likely have been complete Idris-Elba-levels of black due to how he is depicted on statues, etc., but genetic research has corrected that, meaning he was likely lighter-skinned than that, but still olive/dark-skinnedin line with mediteranean/arab populations of today.
No genetics for Jesus exist, but there is no reason to believe he looked any different to the jewish and later muslim population that inhabited and still inhabit the region today, i.e. significantly "darker-skinned" than jewish populations that lived in Europe/Russia for centuries and only (re-)settled in Isreal in the 20th centruy.
