That Rand would be holding the Power at that moment was hardly a Sherlock Holmesian feat of deduction. She has also been seized by Rand with Air before. Elsewhere mention is made that channelers tend to do things the same way all the time, so it could have been as simple as Rand weaving Air at her in the same general direction, though with a different end in mind.
Don't forget also, Rand only saw the results. For all we know she was spinning a whole series of cutting weaves around herself, intending to snap anything Rand tried to touch her with. Or maybe she had some sort of ward around herself intended to show her any place a weave of saidin drew near to her.


He blocked her stuff too, and he was far less experienced. If he could, there's no reason she can't, especially given that RJ has said women have great skill with the weaves.


Interrogating Darkfriends, maybe. Figuring out that the suspicious new leading nobles in each nation were Forsaken would not have been hard given that seems to have figured out their presence from the behavior of the residents. With that in mind, she could have got in touch with her sources in either place (and given her two decade search for the Dragon Reborn, she would almost certainly have taken pains to put eyes and ears in two such cities with significance in end-times prophecies - Illian with its ties to the Horn of Valere, and Tear with Callandor), and gathered enough data to extrapolate which one was which. Though she asks Loial for help with info on Bel'al in Tear, and admits she does not know of the data he offers, the implication is that she DID know all the other things Loial mentions about him, and so could have figured it out from what she did know. And who knows what they might have let slip to her eavesdropping technique? Rahvin gave Mat an awful lot to go on, and Mat wasn't even trying to spy. Moiraine could certainly have used her eavesdropping trick in the same way Egwene did in LoC, albeit with greater skill.

As for the Wise Ones, they're half-wild barbarians. Their dreams might have showed them exactly what season and month the events they foresaw take place in from things like the sun, etc. Maybe the length of the shadows as He Came with the Dawn in their dream-visions was indicative. Maybe they knew that He Who Comes With the Dawn had to show up by a certain date, and they wrote the letter based on that. Don't forget, they were in the Stone for a considerable time - it was early spring when they got there and after midsummer by the time we catch up to them in the beginning of tSR. If the Wise Ones were predicting the date the Stone would fall, they missed by a number of weeks, and the Aielman did not hand over the letter until the last night they spent in Tear. Hell, for all we know, they spoke to the messenger in his dreams and told him to date and seal the letter they had given him!