If an electron is in a superposition of two eigenstates, its wave function is the sum of those two eigenstates. Each eigenstate evolves independently of the other in time. The time dependent wave function has the form
$$\phi(x, t) = \phi(x)\cdot e^{iat}$$
where the $a$ depends on the energy of the eigenstate. Now, what happens when you sum two such wave functions with different $a$ together? Well, they interfere. Wherever both wave functions overlap there will be times when $\frac{\phi_1(x)}{|\phi_1(x)|}\cdot e^{ia_1t} = \frac{\phi_2(x)}{|\phi_2(x)|}\cdot e^{ia_2t}$ (constructive interference), and times when $\frac{\phi_1(x)}{|\phi_1(x)|}\cdot e^{ia_1t} = -\frac{\phi_2(x)}{|\phi_2(x)|}\cdot e^{ia_2t}$ (destructive interference). And that means that the amplitude of the superposition $\phi_1(x)\cdot e^{ia_1t} + \phi_2(x)\cdot e^{ia_2t}$ oscillates with a frequency of $\frac{a_2 - a_1}{2\pi}$.
So, the probability cloud of an electron in a state of superposition is not static. It's oscillating with a fixed frequency that's proportional to the energy difference, and thus actively interacting with the electromagnetic field. The result of this interaction may be that the electron drops into the lower state, or that it gets exited into the upper state. But until it reaches a state without an oscillating probability cloud (usually a pure eigenstate), the electron won't rest until it does.
The preference of the lowest energy eigenstate is only due to our preference for cool environments in experiments: When there's no photon around to be absorbed, the only way out of superposition is to emit a photon. However, there are cases where the electrons prefer a high eigenstate. One such case is lasers: They need to get more electrons into the exited state than there are in the base state (this is called inversion), because that's the prerequisite for the light amplification process. That's quite a bit of science actually, but it happens in every single CD player.
I believe the desire of identifying eigenstates is largely driven by the fact, that it's easy to derive the time dependent wave function once you have your wave function separated into eigenstates: Each eigenstate has its own $e^{iat}$ factor, and that's easy enough to calculate for the entire wave function. And the superposition is also easy enough to calculate. You could simulate the time dependent Schrödinger Equation directly, but that's computationally expensive, error fraught, and imprecise on large timescales. The separation of the wave function into eigenstates allows us to come up with analytical, and thus precise solutions easily.