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?
Answer
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")
Correlations.
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.
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