Double photon emission, half photon absorption Is it possible for atom with two electons to absorb one photon in this manner, 
$$|gg\rangle|n\rangle \to |ee\rangle|n-1\rangle$$ 
or for one electron to emit two photons like this 
$$|e\rangle|n\rangle \to |g\rangle|n+2\rangle$$ 
or maybe two different frequencies like this 
$$|e\rangle|n\rangle_{f_1}|m\rangle_{f_2} \to |g\rangle|n+1\rangle_{f_1}|m+1\rangle_{f_2}$$
I have never learned about this, or thought about this until now, so I assume there's some rule that forbids this that I can't think of. 
 A: The effects you describe are all possible. An example of the first effect is double photoionization in which one photon ionizes two electrons. This is possible because of the correlation between the two electrons and the ratio of the single- and double-photoionization rate is an important parameter to quantify this correlation experimentally and is studied mostly in helium and helium-like atoms.
The second effect is two-photon decay. Although very unlikely, this effect is possible. For instance, the 2s state of the hydrogen atom primarily decays via emission of two electric-dipole photons and this process plays for instance a role in the relaxation of excited hydrogen atoms in the Universe. 
The third process is just sequential de-excitation via some intermediate state. One example would be the production of electronic excited molecular hydrogen in a discharge so that singlet and triplet Rydberg states of H$_2$ are populated. The singlet states will decay by a single photon to the ground state, but the triplet states will first decay to the lowest excited triplet state that is metastable and subsequently decay to the singlet ground state via spin-orbit interaction. 
