Abu- is not uncommon in Sephardic names as well as Islamic. I know several, including a lady who runs a kennel from which I bought a dog; her last name is Abu-..... (I believe from ibin).

Some of the names in the songs come from Turkish languages, much better known in Europe than 'Arabic' or African before the 19th c., and from languages of the Indian subcontinent picked up during the Raj.
I don't really know what you mean by Arabic names (I think you mean Islamic, Muslim) or African names. Many of those in song are mis-used. "Little Black Sambo" was about an East Indian boy, as pointed out above.

Cuff-Cuffin-Cuffee is an old slang word for man, in print from the 16th century in English but probably older; origin unknown. My grandfather called himself an old cuff, and I am one now. Thus from which source the minstrel name is taken is uncertain.

Sambo is rather firmly African; an early American slave was named Sambo, and he made himself known as an escapee (1704, Boston News Letter). There is the old English slang word 'sammy,' meaning foolish or silly, or a silly man, but this is too big a stretch.   

I think you mean Islamic, rather than Arabic, names. Lets not be bigoted (he, he).
In Arabic and Urdu, Abdul means servant; Abdullah is servant of Allah. Abdul appears in Farsi, but not Abdullah. Also appear in Farsi. 'Ala al din means nobility of faith. Abdullah is present in Turkish, but not Abdul. 'Ali is present in all four.
A good survey of Islamic names: http://www.geocities.com/~abdulwahid/muslimarticles/names.html
Islamic Names

I won't comment on African names; too many different peoples involved, too many languages, too many cultures.