do the electron(s) keep going higher the more light you shine (...)?
No, because energy levels are quantized.This means that no matter how many photons you throw at the electron (i.e. the intensity of the light source), it won't jump to an higher energy level unless the frequency ($\nu$) of the photons is right, i.e. if
$$\nu = \frac{\Delta E} h$$
where $\Delta E$ is the energy difference between the energy levels and $h$ is Planck's constant.
Secondly, if you then stop shining light, why will the electrons fall back to a lower level? Will they at all? And why? it seems arbitrary that they will unless acted on by something else.
Yes, they do fall back, and the reasons are two:
- An atom is never truly isolated, and it will interact with the external electromagnetic field.
- Even if we assume that the atom is in free space, far from any source of EM field, it will still be subject to vacuum fluctuations of the EM field and thus eventually decay to a lower energy level. This process, which is called spontaneous emission, cannot be explained if the EM field is treated as a classical object, and its description requires the formalism of quantum field theory. For a more detailed discussion, see for example the Wikipedia page:
Spontaneous transitions were not explainable within the framework of the Schroedinger equation, in which the electronic energy levels were quantized, but the electromagnetic field was not. Given that the eigenstates of an atom are properly diagonalized, the overlap of the wavefunctions between the excited state and the ground state of the atom is zero. Thus, in the absence of a quantized electromagnetic field, the excited state atom cannot decay to the ground state. In order to explain spontaneous transitions, quantum mechanics must be extended to a quantum field theory, wherein the electromagnetic field is quantized at every point in space. The quantum field theory of electrons and electromagnetic fields is known as quantum electrodynamics.
In quantum electrodynamics (or QED), the electromagnetic field has a ground state, the QED vacuum, which can mix with the excited stationary states of the atom. As a result of this interaction, the"stationary state" of the atom is no longer a true eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it. Spontaneous emission in free space depends upon vacuum fluctuations to get started.
If they do fall back, how long does it take till they do?
The probability that the transition has not happened at time $t$ is $1-p$, where $p$ is the probability that it has happened. To calculate the transition probability per unit time, you can use the Einstein coefficients.