You are asking for a description of quantum scattering in the case of Rayleigh scattering, and why the sky is blue.
When a photon interacts with an atom, three things can happen:
elastic scattering (Rayleigh scattering, or for example mirror reflection), in this case the photon keeps its energy level and phase and changes angle
inelastic scattering (heat up material), in this case the photon gives part of its energy to the atom and changes angle
absorption, the photon gives all its energy to the electron/atom system, and the photon ceases to exist
Now Rayleigh scattering, the main cause of the blueness of the sky, is elastic scattering. The photon does not get absorbed, and does not cease to exist. The photon keeps its energy level and changes angle.
You are asking "What happens to the photon and electron during scattering? absorbed and re-emitted? just 'bounces' off etc?"
In the case of Rayleigh scattering, which is elastic scattering, it does not get absorbed, and does not cease to exist. You cannot imagine the photon as a little billiard ball, but in some sense, it just "bounces off" like you say.
You are asking "Why can scattering occur at all frequencies if the energy levels in an atom are discrete?"
Now in the case of absorption, you are correct, the descreteness energy levels (in the atoms/molecules) cause the quantum mechanical phenomenon where we see from experiments, that for a photon to be absorbed, the photon's energy needs to match (or exceed) the energy gap between certain energy levels of the atom/electron system. Now this is not the case for elastic scattering. Now in the case of elastic scattering, the probability of the scattering event is much more dependent upon a relation between the wavelength of the photon and the size of the atom/molecule.
The Rayleigh scattering depends on the wavelength and blue light is scattered most. That means the light we see coming from directions away from the Sun has a spectrum weighted towards the blue.
Why is the sky blue and the sun yellow?
You are asking "what is the quantum explanation for higher frequencies (blue light) being scattered much more (freq^4) than other frequencies?"
And we got to the beautiful QM phenomenon called scattering, where the relationship between the wavelength of the photon and the size of the atom/molecule matters most. In the case of Rayleigh scattering, the atoms/molecules size is much smaller then the wavelength of the photons.
is the predominantly elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the radiation.
The strong wavelength dependence of the scattering (~λ−4) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths.
Now since atoms/molecules' sizes are much smaller then the wavelength of the photons we are talking about (visible in your case), the smaller (shorter) wavelength photons scatter more then the longer, causing the blue color of the sky.