The hunt was for active male channelers, the breach of tower law was stilling them on the spot (or possiblymurdering them outright if black). There was no way to hunt potential male channelers. Female channelers can feel the potential for channeling in a woman with the talent who has not channeled (whether one with "the power inborn" who will spontaneously channel or one who will only channel if guided) and how strong she could become but a female channeler cannot feel a male channeling nor sense the potential for channeling (nor can a male channeler feel the ability in a male or female, only sense if the source is actually embraced in either). Only males who actually channel could be detected and eliminated (and then only by observation of the effects of their channeling or of the taint on their behavior). Cadsuanne had a slight extra edge from a ter'angreal which indicated the direction of nearby male channeling but there is no mention in the series of any weave or device to detect the potential for male channeling other than the testing (which could only be carried out by a male channeler and awoke the ability in the moment of detection).

In the books spanking by the Aiel was for lying (at the request of the person receiving the punishment, to help her meet her Toh for having lied). Aviendha in the show would have been asking her fellow maidens to help her meet Toh for having been captured (which would be considered more shameful that being killed - at least by maidens - Gaul in a similar position in the books does not appear to feel similar shame, but there are stated to be differences in how different societies view things). Similar to how Sulin and the new leading maiden resolved their differences in the run up to Dumai's Wells with a fist fight.

How do you define success? If you mean achieving the dark one's goal of making Rand despair to the point he might choose to do what the dark one wanted (which appears to be ending time during his contest in the bore) then Semirahage due to collaring Rand and nearly forcing him to kill Min (which ironically taught him the command of the dark one's "true power" needed to correctly seal the prison). If you mean achieving their own goals (which vary considerably) then Moghedian (who at least survived and may still get free of the collar). Demandred's main goal was to best Rand (as Lews Therin reborn) in a direct contest, and Rand did not directly face him.

The content of the letter itself explains the apparent lack of urgency - Moiraine explicitly instructs Thom that Mat has to ask about the letter before being shown it. Thom flaunts the letter in front of Mat for a long time before he asks. When re-reading the books you should get a sense of Thom's frustration at the delay which he has to hide from Mat since the best way to get him to ask is to make him think you do no want him to ask.

She would have been in the tower for up to 6 months in The Great Hunt (leaving Fal Dara in late spring for a very rapid trip to the tower and not being tricked into the ways until early autumn). I know it does not feel like it compared to the hunters POV but that is due to the time lapse during their portal stone trip. Then again in the tower for a shorter period (up to a month) in The Dragon Reborn and also thinking of herself as Aes Sedai (or AS affiliated) from being guided by Moiraine in Eye of the World up to being apprentice to the wise ones and then again from traveling to Salidar - overall it feels like long enough to belong, especially as the tower would be attempting to indoctrinate any novice to reduce any attempts to leave.