I will address this:
Am I correct in saying that different atoms are only capable of absorbing certain frequencies/energies
(i.e. different atoms have different resonant frequencies and can only absorb at multiples of those frequencies)?
No. Atoms and molecules are in the quantum mechanical regime, resonances describe classical waves at distances much greater than atomic. Photons come in quantized discrete energy of E=h*nu.
In the quantum mechanical solutions for a single atom, there exist specific energies where an electron can be excited and de-excited from , emitting a photon.
Each is a unique frequency for the atom species, in the above case hydrogen, and that is how astrophysicists identify the elements in the sun and stars, by the spectral lines emitted and absorbed by the atoms.
Note that for higher excited states the energy levels are practically continuous, which allows a large range of frequencies to be emitted and absorbed around the bound state value of , for example, 13.6 ev. It is not sharp, but the photons still carry a discrete energy quantum.
When it comes to molecules, additional energy levels will exist and the energies will be lower than the atomic levels, and the photon energies will be even more diffuse about a central value. For the lattice of molecules in a solid the possible energy states are multiplied by vibrational and rotational levels.
The black body formula approximates the effect of all the possibilities of excitation and de-excitation from energy levels.
Statistical mechanics tells us that the temperature of a gas depends on the degrees of freedom and the average kinetic energy. It gets more complicated with solids, but it is still true that the energy of a solid as defined by its temperature supplies the energy for transitions in lattice and molecular levels that appears as black body radiation. The formula fits approximately, and constants of emissivity and absorptivity are used to fit the data to the theoretical curve.
Edit to address comments ,
The best fitting curve to the theoretical black body curve is the one of cosmic microwave radiation.
Graph of cosmic microwave background spectrum measured by the FIRAS instrument on the COBE, the most precisely measured black body spectrum in nature. The error bars are too small to be seen even in an enlarged image, and it is impossible to distinguish the observed data from the theoretical curve.
One should not be surprised because after all it is a "photon gas"
Matter needs emissivity and absorptivity constants/functions to fit the black body curve, and these are usually found in engineering tables.
For solid matter which has very many possible energy levels and can be very smooth, nevertheless there is a difference between measured radiation to the theoretical black body curve:
Emissivity spectrum of quartz compared to that of a blackbody at the same temperature; quartz emits less energy and therefore has emissivity less than 1.00
The black body curves are useful for estimating the energy output of stars, even though the fit to the black body is approximate, here is the sun spectrum.
Solar irradiance spectrum above atmosphere and at surface. Extreme UV and X-rays are produced (at left of wavelength range shown) but comprise very small amounts of the Sun's total output power)
The differences are due to the collective addition of various energy levels in the plasma of the photosphere of the sun.
Individual spectra are detectable in gases,because the main excitations and deexcitations are from collisions between molecules, and the collisions excite molecular and electronic levels and the return to the ground state is the black body radiation.
For atmospheric gasses, for example, the black body curve is not such a good approximation.
Credit: Data from R. A. Hanel, et al., J. Geophys. Res., 1972, 77, 2829-2841
Nevertheless the black body curve is a good tool for a first order approximation of energy losses, and engineering tables for material emissivity should be used for accurate estimates.