Did the Arab slave trade of Africans, Europeans, et. al. leave significant genetic traces?
Yes. There were identifiable populations of negroes in the northern Ottoman Empire, including an all black village on the northern coast of the Black Sea which survived into Russian imperial rule.
There are clans and tribes among Arabs of the Levant and Mesopotamia (including some of the Bedouins) who are known as "black".
The Arabs don't like to talk about this much; they prefer to forget about the East African slave trade. Also, there is a long tradition of anti-black prejudice in the Middle East, and black presence/ancestry is ignored if possible. (The clans mentioned above are barely acknowledged.)
As to European ancestry: DNA scanning in Morocco has turned up descendants of Icelanders carried off by Barbary pirates several centuries ago.