The phenomenon of observing one entangled particle and noticing the other take on corresponding values... Does this take a finite speed at all or is it instantaneous?
You must distinguish 2 concepts:
Transmitting an information.
The protocol is strict: An observer, at some point of space-time, emits an information (and any information carries energy), and this information is received by an other observer. The speed of transmission of this information cannot excess the maximum speed of information ("speed of light")
Quantum Entanglement is only a special kind of correlations. You may find quantum correlations, but also classical correlations (for instance, in probabilistic problems). A correlation has nothing to do with transmitting an information. It is only a set of numbers, which indicates a joint probability law, of the form $p(a,b)$, which corresponds to a joint measurement of aleatory variables $A$ and $B$
The illusion of "spooky action at a distance"
This illusion appears in the following case. Take a classical or quantum system made of 2 sub-systems, and create a correlation between these 2 sub-systems (so, at a given point of space-time). Then, move these 2 sub-systems so that they are spatially separated. If a measurement is done on A, there is certainly a correlation with the measurement done on B (this is the definition of correlations), but there is no signal transmitted.
It doesn't make sense to discuss the speed of entanglement. It's like discussing the color of honesty or the shape of temperature. Entanglement doesn't propagate information, and there is no cause-and-effect relationship between measurement of the state of particle A and measurement of the state of particle B.
This answer gives a nice analogy involving one glove in the pocket of your coat and its partner left at home. Discovering the left glove in your pocket doesn't cause a signal to go back to your house and cause the other glove to be a right glove.
I am not even sure the following simpler issue has been resolved: is this phenomenon faster than light? To prove experimentally that it is faster than light, one needs to demonstrate loophole-free violations of the Bell inequalities. That has not been done so far, almost 50 years after Bell's work. To prove theoretically that it is faster than light, one proves (within the proof of the Bell theorem) that the Bell inequalities can be violated in standard quantum mechanics (SQM). However, such proof needs using both unitary evolution of SQM and the quantum measurement theory of SQM, and these two are mutually contradictory (the well-known measurement problem of quantum theory).
As for proving experimentally that this phenomenon is instantaneous, N. Gisin proposed some experimental schemes that could prove that, but they have not been implemented. I guess this is even more difficult to demonstrate than the standard loophole-free violations of the Bell inequalities.