Unit tests are never perfect at capturing functionality, particularly in certain parts of an application (such as the GUI), so everyone needs some measure black box testing. Does TDD have anything to say regarding black box testing? If it does not say much, could it be true that while writing unit tests is the developers job, functional black box tests fall into a different domain like that of a business analyst or a dedicated tester?
TDD has evolved to include black box testing, BDD quite specifically uses it as a device (see ATDD).
However, I don't think this answer tells you in whose domain black box testing lies. Personally, I think it lies with everyone. The BA knows what the business wants and Developers and Testers understand edge cases that a BA should never need to understand, often both with completely different perspectives.
And this is why the Gherkin-based test runners have become popular, because they allow BAs, Devs and Testers to write the test cases in the same language as each other (though developers have to put some code behind the text).
You must write the tests first. Therefore, there is no "white box" because the code does not yet exist.
The distinction between black box and white box means essentially nothing in the TDD context.
Yes. Of course.
TDD does not mean "rigidly follow some brain-dead path."
If you want to do additional testing of the GUI, that would be a well-accepted idea.