Do electrons occupy higher energy levels to become excited? Or do they become excited to occupy higher energy levels? What's the causal relationship? In other words, if I had the magical power to extract energy from an electron, what would happen if I forced an electron to emit a photon when lower energy orbitals are all fully occupied? Would the electron simply cease to be? Or would it "rest" on other electrons?
 A: You are using the outdated Bohr model where the electrons were supposed to run around the nucleus as the planets around the sun and only the angular momentum postulated to impose the quantum mechanical demands.
In quantum mechanics, the energy levels belong to the atom as a whole, and it is the atom that changes energy levels, the position of the electrons being probabilistic and mathematically described by the wavefunction. You cannot extract energy from the electrons of the atom, but from the atom as a whole. They are orbitals , not orbits.
A: 
what would happen if I forced an electron to emit a photon when lower energy orbitals are all fully occupied? Would the electron simply cease to be? Or would it "rest" on other electrons?

We say that electron is excited when it occupies a higher energy level. Electron from a lower energy level becomes excited after receiving energy (e.g., in a form of photon), leaving behind a vacancy (in semiconductors such vacancy is called hole). Electron then may return to this vacant place by emitting a photon. Sometimes electrons rearrange, so a vacant place might be elsewhere, but it is still present. If all the lower states are filled, and electron nowhere to go by releasing energy, then we cannot really call it excited, and, since it has nowhere to go, it cannot emit a photon.
A: "Becoming excited" is the same as "occupying a higher energy level". There is no difference between the terms and therefore no causal relationship because they're the same thing.
A: Technically, “Becoming excited” and “occupying a higher energy level” is the same thing…And electrons can be considered and waves too right? How can waves “rest” on each other, that sounds weird.
It’s not possible to force an electron to emit a photon, it has to be pushed to the next energy level and it has to come back down to the 0-state, in the process of returning (which happens in nanoseconds)it emits a photon
I feel that extracting energy might be against the laws of conservation
A: TL;DR The electrons around an atomic nucleus try to place themselves as close as possible around the nucleus. This happens because the negatively charged electrons are attracted to the positively charged nucleus. If the atom is an ion, the atom will adjust its number of electrons back to the number of protons at the next available opportunity.
However, a stable state of equilibrium does not exist and this is related to the exchange of energy with the environment. Above 0 Kelvin, the electrons are never in the ground state and when we talk about ground states of electrons, we mean our observations at room temperature. TL;DR
At room temperature, electrons are constantly exchanging energy packets. If you change the temperature equilibrium and disturb the atom with extra photons, then the excited electron moves away from the nucleus. The electrical attraction between the proton and electron is the dominant interaction between the two. So adding energy changes this interaction. The absorption of photons increases the distance, the subsequent emission decreases the distance again.

do electrons become excited to occupy higher energy levels? What's the causal relationship?

The electrons absorb energy and therefore take up higher energy levels in the atom or even leave the atom.
