What are the oaths the seanchan think the westlanders broke or abandoned?  Having a hard time finding it.

I don't think this is ever really explained.  We know that some people (the Watchers over the Waves) still believed that Hawkwing's armies would return from across the Aryth Ocean.  Perhaps Hawkwing made everyone swear to accept his heir as his successor, whom he then sent across the ocean and whom most people forgot when the Hundred Years War came up.  Whatever the truth, it's pretty clear that the Seanchan (probably unintentionally) distorted the truth to fit with their own cultural bias, and consequently had wholly unreasonable expectations when they came back to the Westlands.

Is Beslan gay?

 

I know this sounds silly but I was re-reading Winter's Heart and when Beslan is talking to Matt about helping him leave he says:

 

"My mother won't be pleased if she learns I am helping her pretty leave Ebou Dar, Matt.  She will marry me to someone with a squint and a mustachelike a Taraboner foot soldier."

 

I was thrown a little, unless he simply meant that she would find some woman with facial hair and a squint.  But it occurred to me that he might have been talking about a guy!!

I think the idea was that she would marry him to an extremely ugly woman, and Beslan phrased it this way because he finds facial hair on a woman highly unattractive.  As far as I know, there are no references to male homosexuality in the WoT at all; more importantly, even if there were, I very much doubt that any WoT society would use the term marriage for a homosexual relationship.

My impression was that the Guardian detects channeling within some radius, much larger than the no-Source "bubble;" it's not supposed to need to detect channeling inside the bubble.  Every time it detects nearby channeling, the Far Madding people triangulate to figure out exactly where the channeling occurred.  When Nyneave channeled, they were astonished to realize that the location of the channeling was actually inside the city.

Incidentally, the Guardian does not seem to stop channeling as such; rather, it stops channelers from embracing the Source (unless they've brought their own supply).  Nyneave considers this when she sees Rand about to rain down fire on the city in TGS, and concludes (tentatively) that Rand would have no trouble weaving into Far Madding if he himself were standing out of range, and thus able to embrace the Source.

At the beginning of the chapter "The Tower Stands," Egwene thinks

Siuan had found her Great Serpent ring; it felt very good to have it on her finger again.

As best I recall, the last we saw of Egwene's ring was in KoD, when Silviana took it for safekeeping, "until you have a right to it."  How did Siuan get it?  Or is this a mistake?

"The Tower has been a bulwark against the Shadow for three thousand years.  It has guided rulers to wise decisions, stopped wars before they began, halted wars that did begin.  That humankind even remembers that the Dark One waits to escape, that the Last Battle will come, is because of the Tower.  The Tower, whole and united.  I could almost wish that every sister had sworn to Elaida, whatever happened to Siuan."--Moiraine, TFoH, Chapter 15 "What Can Be Learned in Dreams"

Ironically enough, Moiraine has here relied mostly on the role of the Gray and Brown Ajahs to justify the Tower's existence.  The Red and Yellow Ajahs are the ones that, in my opinion, have done the most obviously worthwhile tasks (however unlikable the Red may be).  Certainly, we may question the extent to which the Aes Sedai guidance of rulers was actually "wise," as Moiraine claims.  However, one important point here is that the Aes Sedai--even Black Ajah, given that they will have to answer to other Aes Sedai--will almost always be better than the worst of rulers, who are totally and overtly abusing their power for personal gain.  [People like Galldrian, or some of the High Lords of Tear.]  Additionally, Aes Sedai have been through an education process which, in theory, ought to have improved them and also weeded out any idiots.

Another important role the Aes Sedai have fulfilled is to keep other groups of channelers in check, for instance, preventing the sort of chaos that happened in pre-Hawkwing Seanchan with the Armies of the Night.  Their biggest mistake, as so many have pointed out on this thread, was to come to believe their own propaganda.