Boolean Circuit in a Black Box?
Actually, the problems BBB and BBB-F above are not currently specified as languages or decision problems (the black box is not explicitly given to us as a binary string of some kind, is it?), so these problem cannot be in NP, PP, PSPACE, or even decidable/undecidable. A fundamental property of languages in the computability/complexity sense is that no bit of the initial input is "hidden" from you.

As suggested by M. Alaggan, the problems BBB and BBB-F are probably closest to the "learning with membership queries" framework, where one can only access a target function by querying it at various points and tries to infer the function. Nevertheless, both problems require superpolynomially many queries to the black box in the worst case, so they are both "intractable" in that sense. For BBB and BBB-F, your intuition is correct: it requires $\Omega(2^n)$ queries just to check that $B$ is not the "all-zeroes" function. 

Even if you assume that the black box in these problems can be indeed modeled by some polynomial size circuit, it is highly unlikely that this helps. For example, if pseudorandom functions exist (a common cryptographic assumption) then for all $n$ there are polynomial size circuits $C_n$ with $n$ bits of input that cannot be distinguished from a completely random $n$-bit function, by any probabilistic subexponential time algorithm with black-box access to $C_n$. It follows that no subexponential time algorithm can efficiently reconstruct $C_n$ just from queries to $C_n$.