Black body is an object, hence it's made of atoms.
No, a black body is not made of atoms. It is an imaginary object which has the special (but physically unrealizable) property that it is a perfect absorber - any incident electromagnetic radiation is absorbed, with zero reflection and zero transmission. Under certain conditions, many objects in nature (stars, rocks, people) can be modeled as black bodies (or some modification thereof) to varying degrees of accuracy.
Depending on what atom it consists of, that's how the emitted spectrum should be in my opinion. If it contains hydrogen, absorbed light on black body would excite electrons and electrons of hydrogen can only produce certain wavelengths.
The electromagnetic radiation emitted by an object is not determined exclusively by the electronic transitions which are possible in its constituent atoms. Solid objects have electronic energy levels which form broad, continuous bands, not discrete spectral lines. Stars are comprised of plasma, not neutral atoms; photons are produced via the scattering of charged particles (bremsstrahlung) and their energies are further randomized via Compton scattering.
That means when we talk about perfect black body, we shouldn't talk about atoms inside it as there's no atom existing that absorbs or emits light at any wavelength.
Yes, that's right.