Another key quote/Q&A from Brandon:

Q. Was the Compulsion which Elza told Semirhage about, Verin's work from after Dumai's Wells, and was it Shaidar Haran who told her about it and to ask Semirhage to remove it?

A. Yes.

So, we know for a fact that it was SH who told Elza about her Compulsion, AND told her to have Semirhage remove it.

If Graendal was involved in this plot at all (e.g. it was Graendal who delved Elza and told SH about the Compulsion), SH would have had Graendal remove the Complusion right then and there, rather than tell Elza to go have Semirhage remove it. Especially since Graendal is the most expert on Compulsion of all the Forsaken, so we're lead to believe.

The only real mystery in all this, it seems to me, is how SH knew/determined that Elza had been Compelled.

My guess is, SH has a special ability to ... evaluate ... the mental condition (amongst other things) of those who've sworn to the DO. Here's another telling summary of what Brandon said about SH:

This lead to lots of discussion about swearing to the Shadow – basically, it’s a very bad idea and you forfeit some very basic protections when you do. Shaidar Haran has special power over those that swear to the Dark One, and the Forsaken in particular. I asked about Alviarin’s special mark, and he said Shaidar Haran has similar power over her. The implication is that there are lots of ramifications for swearing to the Dark One. Brandon mentioned that this makes Verin all the more remarkable.

I guess that "special power" enabled SH to determine that Elza had been Compelled. But, he couldn't do anything about it himself - he had to have Semirhage remove it.

(Edit: There's a second mystery too - how Elza learned those special AoL weaves. And, I suspect SH was responsible for that, too. Not by showing her weaves - he can't channel saidar, obviously. But rather by implanting how to do it in her mind, much like the 'Finn implanted memories (and associated skills) in Mat's mind. SH could have retrieved the information he implanted in Elza from one of the other female Forsaken - Cyndane or Moggy, perhaps.

This explanation would help explain Brandon's quote saying that "Elza had been given knowledge of several rarely known weaves, and in other ways made into a tool of Shaidar Haran. Not all of it was pleasant for her.". I would guess this "memory implant" (if that's what really happened) was not particular pleasant. Plus, note the wording "given knowledge of", rather than "taught". That fits pretty well with what I suggested.

These ideas make SH rather powerful, I admit, but not much more than the 'Finn, really.)