If you wish to communicate so that what is understood by the people to whom you wish to communicate is what you mean, then I urge you never to think of relying on formal logic, however unambiguous it might seem to you.
So the problem with "Is he black or male?" is not just ambiguity in the ears of the listeners (because most listeners will not be aware of the different types of 'or') but is that the question is bizarre. You have to spell out what you mean.
I share Jason Bassford's hatred of 'and/or' on stylistic grounds, but there is another objection: a question with 'and/or' in it is highly likely to be perceived as confusing, particularly in speech. Complicated logical questions do need to be posed clearly, and that means with repetition and redundancy.