Is the movement of electrons truly random? The result of rolling dice is considered pseudo-random because it depends on an almost endless list of factors (how you roll it, the terrain it lands on, etc.), but it is not TRULY random. Is the movement of electrons TRULY random or just pseudo-random? If it is just pseudo-random, what factors would typically effect it (gravity, temperature, etc.)?
 A: In it's current state, it's difficult (maybe impossible is a better word) to answer your question precisely because, among many other things, your definitions of the terms "random" and "pseudo-random" are not precise.  It's also difficult to answer your question because electrons don't "move" in the sense that objects are modeled to move along trajectories in classical mechanics.  
Here are some facts that may allow you to answer your own question since you probably understand what you mean by random better than your descriptions let on:
Consider the electron in the hydrogen atom.  If one specifies the quantum state of the electron, then this will tell you certain probabilities that if a measurement is made on the electron, then one will measure specified values of observable parameters like it's position, or its velocity.  The fact that outcomes of such measurements are probabilistic is not a result of an ability of humans to construct precise measurement procedures and apparatuses, it's an intrinsic property of nature.  You might even be inclined to use the term "random" to describe the outcomes of such measurements, but I think that would be misleading to most people.  "Probabilistic" would probably be a better term.
Most importantly, let me emphasize that you can't escape the probabilistic nature of the measurements of physical quantities in quantum mechanics; it's simply a fact of nature that the state of a quantum system can only give these probabilities.  We're not just "missing information" about the state of such systems that's somehow "hidden" and would specify the state of the system more sharply.
A: The result of rolling dice is not pseudo-random because the dice is interacting with surrounds, and this is a LPS with Poincaré resonances. As a consequence of the existence of Poincaré resonances, the classical system is unpredictable even if the initial state is known with complete precision.
The same about electrons, except that their unpredictability is more important due to their small size when compared with surrounds, baths, measurement apparatus... of course, electrons are quantum particles and follow quantum laws:
"Quantum Theory of Non-integrable Systems" T. Petrosky, I. Prigogine and S. Tasaki Physica A 173, 175-242
"Extension of scattering theory for finite times: three-body scattering" T. Petrosky, G. Ordonez and T. Miyasaka Phys. Rev. A 53, 4075-4103 
