Can one stimulate emission of a photon with an energy different from the emitted photon? Suppose I have a three-level system with $E_0$ the ground level, $E_1$ the intermediate and $E_2$ the upper level. In thermal equilibrium they will have a certain probability distribution according to the Boltzmann Statistic, in a laser one needs a population inversion, but that doesn't matter for my question. 
My question is this: 3-level laser uses the $E_1 \rightarrow E_0$ transition because the $E_2$ level decays quickly (by design), i.e., emission of an $E_1$ photon is stimulated with an $E_1$ photon. But is there also a stimulated emission of energy $E_1$ for an incident photon of energy $E_2-E_1$? My thinking is that a photon energy of $E_2-E_1$ kicks the electron from $E_1$ in the upper state $E_2$, which will then decay, and every once in a while it should decay into the ground level, not back into the $E_1$ state. Is that correct? And if that is so, is the momentum of the emitted photon with energy $E_2$ aligned with the momentum of the incident photon of energy $E_2-E_1$?
 A: A photon of energy $E_2-E_1$ can cause stimulated emission from the state at $E_2$ to the state at $E_1$.  It cannot cause stimulated emission from the state at $E_2$ to the ground state.  The photon produced in stimulated emission has the same energy and momentum as the stimulating photon.
Photons of energy $E_2-E_1$ can stimulate various nonlinear effects,  such as two-photon absorption from the ground state to the level at $E_1$.   But it can't induce the direct transition that you imagine.  
A: WIMP you wrote "A particle in $E_1$ can absorb a photon of energy $E_2-E_1$, right?" - Yes
"That photon can subsequently decay into $E_0$, not only into $E_2$, right?" - No, it will most likely decay into $E_1$, then decay to $E_0$.
"Depending on how much $E_0$ is populated, this will result in an energy gain, right?" - The decay to $E_0$ is an energy loss equal the the energy of the photon emitted. If all the particles are in $E_0$ then they have zero (point) energy.
"Alas, I suspect the so emitted photons are not correlated. I am asking for something that shows this suspicion is either wrong or right." - If the decay is caused by stimulated emission then the emitted photon will be coherent with the stimulated photon. If the emitted photon is produced by spontaneous emission then it will be emitted in any direction.
