Most importantly, notice that at the first meeting with Cadsuane upon their arrival, Bair and Sorilea already knew Cadsuane had male/female a'dam and took the initiative to ask her about that. Cadsuane wasn't in any way surprised that they knew. They knew she was the one keeping them. So, it wasn't a secret that Cadsuane had the DM.
That she had it was probably fairly well known; WHERE she had it (that it was even in her room rather than some other hiding place) almost certainly wasn't. Otherwise why would she go to such great lengths to conceal it?

There's no indication anyone DID search her jewelry box--or anything else. When Rand confronts her with the box that had held the adam, it's that object and nothing else. As I noted above, this wasn't a case of someone ripping apart her room because it HAD to be there somewhere: Evidently, whoever obtained it knew exactly where to go and how to get it with a minimum of effort (which, given all the inverted weaves, again indicates either Sorilea or Bair, and the former is more likely in light of the points made by Lord Haart and darius_sedai. )


I wouldn't call that either "obvious" OR "fact. " I'm always leery of the logic that says, "this isn't the case because [semi-major character x] hasn't wondered about it. " Gawyn hasn't wondered whether Egwene will make him her Warder, or marry her--except he almost certainly has, we just haven't been privy to it. We've never seen Elaida wonder how Rand will defeat the DO and reseal the Bore, so she must never have considered it. Of course, where I encountered this reasoning most was in the debates over whether Verin or Sheriam were Black Ajah: OBVIOUSLY they COULDN'T be or something in their POVs would've revealed it--except that those parts of their POVs that WOULD'VE revealed it were deliberately withheld because Jordan didn't want to tip his hand. The AUTHOR is omniscient, and sometimes the narrator; if the READER were we wouldn't need to RAFO.

I don't believe we "know" any such thing. It's a possibility, but I don't think the rationale you outline there a very STRONG possibility, in part for reasons others have already offered: I don't think Elza was allowed use of the TP (she's certainly never proven herself to the degree I'd expect that to require, unless getting herself captured count) and even is she had it's doubtful anyone could "teach" her how to weave flows with it because it's undetectable by others. Apparently, Semi couldn't detect Rand using it even when they both wore the adam: She figured it out by process of elimination when she realized he couldn't have used the One Power. I also don't think it's as simple as "look for the thing with a saidar aura" because that probably covers a lot of ground in Caddys room (or any Aes Sedais) and there's no indication the place was dismantled (just the opposite, in fact. ) Plus I don't think it would be easier to unravel saidar weaves with the TP than it would be to do so with saidin.