Does basic QM allow for superluminal "particle movement" during wavefunction collapse? Can particles move superluminally away from their "expected values" using basic quantum theory? 
Here's an example: The eigenstates of a harmonic oscillator are defined from $(-\infty, \infty)$. This means there's a nonzero chance of measuring it at any arbitrarily far distance (at exponentially small, but albeit nonzero values). 
It seems as though nothing stops a wavefunction from collapsing arbitrarily far away from its expected value.
For example, for the groundstate of a harmonic oscillator, the wavefunction is:
$$\psi_0(x) = C{e^{-x}}^2$$ with an expected value of $\langle x \rangle = 0$. When this wavefunction is collapsed, there's a probability of it "jumping" away from its expected value an arbitrarily far distance. 
(As a side-note, things don't make a ton of sense trying to get a hold of it simply: 
If we crudely define "speed" as $\frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$... it seems as though if $\langle x \rangle$ is used instead of $x_i$ then basically any nonzero value that our system collapses to will cause some superluminal jump [from the point where the wavefunction was still uncollapsed and had an expected value of zero]. ) 
You might argue that using the expected value as the particles location is the "mistake" here and that I (to identify superluminal travel) must make two separate measurements separated in time. But intuitively, the location of an electron in a groundstate is very, very certain within a certain area. You would think that if it appeared to 'jump' very far away from that region that this would count as some sort of "motion". 
Consider this example:
What if I have a grid of independent electrons all in groundstates. I can individually prepare information in the z-components of their spin (spin up or down), and I can do so without disturbing their wavefunction in X (Note: I'm talking about position, not the x-component of spin, which is an independent degree of freedom).  Immediately after I prepare these states, I make a measurement of position for all of the electrons. Each electron has a super small, but finite probability of collapsing very far away (say the moon), for example. So there is a nonzero chance that my entire encoded information travels superluminally to a target destination. It's small, sure, but it's nonzero.  
This suggests to me that casuality is only preserved probabilistically, (The average can't violate causality, but individual events can.) but seems a little bit too 'out there' to be true. 
 A: No, there is no superluminal movement.
The "wave function collapse" is not a "movement". To define a movement, you would first need an initial position and a final position - i.e. two fixed real numbers (or real vectors), and then to define its speed you need moreover the path taken between them to get the distance, and divide by the time elapsed from the point at which the particle departs the initial position to the point it arrives at the final position.
There are many different quantum interpretations, of what constitutes the precise physical meaning (or not) to the wave function, and they might all understand this in different ways and all of them are, as far as we can tell, equivalent. However, the key here is - which is independent of interpretation - that in the collapse you do not have a movement as just defined. There is simply not a real-number value that we can say as the initial position, even if the final position can be more "confidently" assigned a value as it in this case it is obtained through a measurement. The expectation value is not a position you can say the particle is actually at any more than any other possible value (though this again may depend on your interpretation - if you use Bohmian mechanics, then there are "hidden" positions "underneath" the wave functions, but I'm saying with the common mathematical framework), it's just the average that many repeated measurements on many copies of that quantum state will produce. Thus its change from one value to another, likewise, does not constitute a motion. You can't measure at the initial, wait, and then measure again because measurements change things in proportion to the amount of information they extract from the system. After you did that first measurement, your second one won't even be statistically the same anymore as the original scenario. The only way to measure a quantum system's physical parameters like position and not disturb it is to extract zero information, which is, as one can imagine, rather useless and would not allow you to define any notion of "motion".
Causality is not violated. To properly treat causality, you need to upgrade from usual particle-and-wave QM to relativistic quantum field theory, because the former is essentially formulated on the background of Newtonian absolute space and time, the latter is the proper Einsteinian view. In this case, quantum particles become energized states of a quantum field (thus how that "mass can be a form of energy"), and measurements are performed on the field. Thus each point in space gets its own quantum operator representing a field measurement (like holding a voltmeter between two points) at that point in space. Wave function collapse occurs to the field as a whole. The operators representing the field at space-like - i.e. "faster than light"-reachable, or causality-violating - positions commute, meaning that collapse of one does not lead to collapse in the other. (Namely the field wave functions for each operator only change phase, which is unobservable, but not magnitude.) The collapses occur within a "meta" probability space of field values, not physical space (e.g. a probability distribution of voltages you can measure with your voltmeter when it's connected between two points.).
Conversely, from the viewpoint of basic particle-and-wave QM, there is no violation of causality either. Basic QM assumes Newtonian absolute space and time. There is no speed limit of causality any more than in classical mechanics. Neither theory predicts causality violations. It's just that the real-life Universe is not based on Newtonian absolute space and time, and we can only treat it as being such under suitably limited circumstances. Using the instant collapse of spatial wave function in particle-and-wave QM to argue for a causality violation is actually a fallacy of equivocation, equivocating on the meaning of space and time. You're having it play the role of both Newtonian and Einsteinian space-time at once.
The problem of causality vs. relativity in QM is a done problem thanks to this and thanks to QFT, which is the correct framework, in terms of what we have available, to address this problem since it fundamentally is a relativity-related one and thus you need a proper relativistic version of QM to make any sense out of it. QFT is that relativistic version. It's only in interpretations where one tries to assign some hidden classical states "underneath", like Bohmian mechanics, where it might be a problem. (Generalizations of BM to QFT are significantly non-unique and subject to considerable dispute, and QFT, not QM, is the more fundamental theory.) Even then, posited causality violations would be unobservable because the mathematics that actually describes what you can observe is unchanged between interpretations. ("Interpretations" that do change it are not, strictly speaking, interpretations. They are separate physical theories.)
A: 
Can particles move superluminally away from their "expected values" using basic quantum theory? It seems as though nothing stops a wavefunction from collapsing arbitrarily far away from its expected value. [...] This suggests to me that casuality is only preserved probabilistically.

Yes, in the basic QM taught in a first course, particles can move faster than light. That is, wavefunctions can spread arbitrarily faster than light, and it's possible to detect a particle on Andromeda one second after it's detected on Earth, and these effects do not cancel out probabilistically. 
This is, of course, not in contradiction with relativity because basic QM is explicitly nonrelativistic. It knows absolutely nothing about the speed of light. You only have causality in the theory of relativistic quantum mechanics, also known as quantum field theory. 
Unfortunately, this point is muddled because of the famous EPR thought experiment, which uses entangled spins. Since spins and spin measurements behave the same way in both nonrelativistic quantum mechanics (NRQM) and quantum field theory, one often treats the system in NRQM for simplicity. One can then prove that no information can be transmitted faster than light in NRQM using spin measurements alone. (In fact, no information at all can be transmitted by spin measurements, faster or slower than $c$, because NRQM doesn't know what $c$ is.)
Many textbooks and all popular books then oversimplify this to "no information can be transmitted faster than light in NRQM", but this is absolutely false. 
