Elaida tried to have it proven that Alviarin was secretly communicating with the Dragon? How did she know about that?

It was more like a guess, than real knowledge. It's not an unreasonable guess either, given Elaida saw first hand what Alviarin was doing--intentionally disrupting the cohesion of the Tower, which would serve either the Shadow or the Dragon (note: Elaida wavered back and forth on believing there were Darkfriends in the Tower, and seemingly decided there weren't after Siuan insisted there were. Given that Alviarin's attempts seem to serve the Dragon).

 

Finally when you consider the way Alviarin took charge of Elaida, the idea that Alviarin might not have made her own moves towards Rand seem unlikely. It must also be remembered that in itself, Elaida wasn't just trying to find out if Alviarin had contacted Rand, she was looking for something damning with which to sever Alviarin's control over her--which in the end came instead from Alviarin's long absense during the time when the Rebels appeared.

Beonin was a turncoat, it is possible, that through Egweene, it was revealed that a letter existed. She revealed the attempt at the gates, so why not that as well.

 

I'm grabbing at straws here.