>Where two particles interact regardless of the physical distance between them.

Technically this is wrong - the two particles are pair-linked by state no matter the distance between them. They don't send information back and forth. We don't know the mechanism for preserving the entanglement.

>Entanglement has also been described as a measurement of one particle that decides the properties of another because the interaction between them determines their shared properties that must be conserved.

This is closer because at the most fundamental level is when two particles are entangled, the shared state constraints between them are preserved until measurement of either or both (and possibly after). But the measurement simultaneously collapses the state of both particles, potentially even when done to the original system state before the particles left it. However, again, we don't know the underlying mechanism (is it non-local binding? pilot waves? collapse of probabilities?).

The wrong way to think about entanglement is a pair of particles that send info back and forth between them to make sure everything checks out - they share state, they don't exchange it. This is why parallel worlds, pilot waves, etc. are theories about how the state is shared are more accurate because they attempt to preserve the state constraints globally. Nothing about entanglement moves faster than the speed of light.

And to answer your question, yes, entanglement does appear to be transitive to other particles under the right conditions, so you can chain state constraints.

This assumes that entanglement is correctly understood by experiment today, which is still not entirely clear, even though the physics community has settled on Copenhagen for the most part.