"Han vil vite om jeg er spansk."
Translation:He wants to know whether I am Spanish.
In English, we would likely say "He wants to know if I am Spanish" or, "He wants to know whether or not I am Spanish." The question is sort of an incomplete sentence the way it is.
There's possibly some dialectal variation in English, but "He wants to know whether I am Spanish" is perfectly fine to me (and your two alternatives both sound odd)
I agree; when using the word whether in this way it is usually paired with "or not." And "if" is the more likely word to be used for such an inquiry. I'm an American who's lived in many regions of the USA.