I answer that one for now since my thoughts are pretty straightforward on that one.

I think Verin meant to use Alanna's ability to find Rand. Alanna's the one Verin could send on a mission anywhere in the world from which she can return to Rand on short notice and without any trouble. She's ideal to give a timely warning to Rand about... something Verin thought important Rand received a timely warning about. She would make an ideal spy, for instance.

I suspect she may also have forewarned Alanna the Shadow knows about her bond (for Min, Verin can only suspect and thus would think it's a saidin bond invented by Rand and wouldn't assume too much about its nature; Elayne's and Aviendha's bonds are almost certainly below her radar) and she has to stay low and be ready to release the bond if she has the slightest suspicion she's about to get caught. Verin may also have been aware of plans of the Shadow for Alanna. Alanna would be the logical choice for this, since she was with Verin in the TR when Luc/Isam got involved, and thus was the one Verin had within her reach who once she found Lan can spot any attempt by Slayer to deceive Lan using his Luc appearance, which for having seen it in action with Perrin, Verin knows is one of Slayer's tactics. It makes all the more sense when you consider the DP revealed to Verin Luc and Isam are mysteryously tied and one is alive and the other is dead but both are, that she heard the Shadospawn used Isam has their battlecry but has not seen the slightest evidence of Isam in the flesh. Perhaps Verin wrongly deduced the DO simply transmigrated Isam's soul in Luc's body. We know Slayer can have either appearance, but unless she's got dealings with Slayer we don't know about - certainly a possibility given Slayer said a few non-Forsaken know how to find him but wouldn't dare give him orders, Verin doesn't know he has two appearances).

Any of that would explain why Alanna disappeared and was careful not to give any hint.

Another possibility that comes to mind is that Verin sent her to find Lan and forewarn him to expect the Shadow will send Isam Mandragoran against him at some point. The shock it's Isam leading the army he's facing might be very bad for Lan. OTOH, unless she spied on Nynaeve and Lan, Verin shouldn't really know where Lan went and left so soon after she had little time for anything beyond noticing he was gone, so if that's the mission she gave Alanna, she deduced Lan went North but Alanna will likely wander around to find him, especially that she couldn't anticipate Nynaeve dropped him at World's End.

Another possibility is that Verin sent Alanna on a crazy dangerous mission, such as attempting to locate Moridin's fortress near SG (as it's been confirmed by Maria Verin was at the DF social, she knows that fortress exists), or spy on activity in Takan'dar.

A forsaken Verin may very well have been involved with, because she was so well placed near Rand for quite a while, is Demandred.

But there's really a ton of possibilities. I'm just very partial to the notion Verin chose Alanna for something specific because she can Travel and can return to Rand in a timely manner.

I don't think Alanna's bond will be very relevant otherwise, unless she gets captured. I think Rand's redundant bonds protect him from the worse of the effects Alanna's death would have, but the Shadow doesn't know this, so if they plan to kill Alanna at a specific time, their little plan might fail..

I get the feeling Alanna's will turn out to be Verin's most relevant letter. We know we're gonna see more ramifications of Verin's plans in AMOL, but RJ can't have overdone it either. Already, he made her fail with Mat.

The letter Galad has is perhaps the most puzzling of them all, but I'm not 100% convinced it's a letter from Verin. It had a red seal. He appears to have gotten it in Whitebridge, of all places. He was coming back from fetching Berelain from her entourage when we saw him put it in his pocket. Annoura? It would be hilarious if this turned out to simply be a smithen love letter Berelain just gave him, however. That could fit with either Sanderson's or Jordan's styles of humour.