Is it probable for particles to become entangled under natural conditions? Under what circumstances could particles become entangled without our intervention? How frequent would that have happened in the various stages of cosmic expansion?
Is it possible that particles may have been entangled in early cosmic history and remain so today?
If the answer to the third question is 'Yes', would not the search for the modulation of these early entangled particles be of tremendous value?
Consider that particles were very close together in early cosmic development. If those entangled particles are now hundreds of light years apart, and have other particles that are also entangled (group entanglement), would those particles not be a means to detect the communications/modulations of other species who have also discovered the entanglement phenomenon and are subsequently 'listening across the cosmos' for those modulations as well?
 A: Yes, particles become entangled without our intervention all the time. However, it isn't relevant because of the monogamy of entanglement. What monogamy of entanglement means (intuitively) is that a particle can have a large amount of entanglement with one, or maybe two, other particles, but it can only have a little bit of entanglement with each of many other particles. 
So if two particles were entangled in early cosmic history, then in order to retain their entanglement today, they need to have not interacted strongly with anything else in the meantime.1 And even if you had an entangled particle, and you knew it was entangled with some other particle somewhere in the galaxy, it doesn't help you do anything unless you can identify this other particle. Even worse, if a particle has a small amount of entanglement with millions of other particles, you need to identify a large number of these particles before you can do anything with this entanglement. 
1Theoretically, a particle can interact with another particle and retain its entanglement with the first one (as long as it doesn't gain any entanglement with the second). Unless the interaction with the second is very weak, this seems rather unlikely to me.
A: 
Under what circumstances could particles become entangled without our intervention? How frequent would that have happened in the various stages of cosmic expansion?
Is it possible that particles may have been entangled in early cosmic history and remain so today?

The fact that two particles interacted at some point in the past doesn't mean they are entangled now in the sense that you could get a violation of the Bell inequalities. Each particle will in general have interacted with lots of other stuff, which means the information necessary to produce the violation will be scattered in the stuff it has interacted with.

If the answer to the third question is 'Yes', would not the search for the modulation of these early entangled particles be of tremendous value?
Consider that particles were very close together in early cosmic development. If those entangled particles are now hundreds of light years apart, and have other particles that are also entangled (group entanglement), would those particles not be a means to detect the communications/modulations of other species who have also discovered the entanglement phenomenon and are subsequently 'listening across the cosmos' for those modulations as well?

No. The entanglement would not be a means of communication across vast distances. Interacting with one half of an entangled pair has no effect on the other member of the entangled pair.
A: I'm afraid that the recent emphasis on quantum information theory has muddied the waters for many about what entanglement really is.  Entanglement is any circumstance where two or more spatially indistinguishable particles interact and either maintain that interaction or spatially separate and fail to interact with other quantum systems.  Count the number of atoms in the universe and you will have the approximate number of entanglements that have occurred without our intervention. Bound electrons (note the plural) in an atom are entangled as are the unbound entities measured by Bob and Alice.  
