It really has nothing to do with potential energy; for instance, if you entangle the polarizations of two photons the energy transport occurs independent of the results of your polarization measurement.
You're more on the right track where you say, "the particles have no definitive state within the universe, as their possibilities haven't collapsed." It's not exactly about whether the particles have separate identities since you can also entangle distinguishable particles. If the particles are not entangled, one can measure some property of each and the results will be independent of one another. But when they are entangled, when you measure one particle that changes the possible outcomes of a measurement on the other - instantly.
You asked, "what would happen if you moved the particles apart at close to the speed of light and then forced one of them to "pick" a state." This is an experiment done many, many times on photons, which of course move apart at the speed of light. In the experiment one measures a property (frequently polarization) of one, which in effect forces it to pick a state. The wavefunction changes from the entangled state to one where the one you measured is in a pure state corresponding to the result of the measurement, and the other one is now in whatever state the measurement of its partner would dictate. What makes this so interesting is that there is pretty much no way to maintain that the entangled wave function is merely a measure of our ignorance. (In other words, suppose I had a red marble and a blue marble and placed them in identical boxes. You fly to Tokyo and I fly to London and when you open your box and see a red marble, you know I must have a blue marble in my box. The situation with photons is fundamentally different from that. If someone "prepared" color-entangled quantum marbles they would be neither red nor blue until one box was subject to a "measurement" (one of us looking, or really, when any physical process occurs that would depend on the marble having a definite color; consciousness really is not necessary for quantum "measurement"). There is a mathematical test of the difference known as violation of a
Bell inequality, and many tests of Bell inequalities on entangled photons support the indeterminate state picture and not the "well, it was really blue all along, we just didn't know until we looked" model.)
As far as re-entangled them, you write, "Simply existing in a specific state in a specific space-time has then given each of them a separate identity, and they must again exist in a minimally proximal space-time region to have their identities "re-confused" to the point where they can be coupled again?" That's pretty close. You don't absolutely need to bring them together again. But they will at least need to interact with something that has a common source; imagine two atoms widely-separated that each interact with one photon of an entangled pair; something like this could re-entangle them. "Remote state preparation" is an active area of research, as is "teleportation" (and sometimes their definitions blur). And again, it's not strictly speaking about particle identity. Mathematically, entanglement is about the form the wavefunction for the entire system of particles can take. If you can write it as a wavefunction for one particle times the wavefunction for the second particle... times the wavefunction of the Nth particle, the particles are not entangled. The particles can be very different - say, a photon and some atom - yet have a joint wavefunction that cannot be written as such a product.