There may be some posts on the last page I missed, and I should probably go back and read those. Still, I think I can respond to your newest post.
The door terangreal does require channeling to operate, but the likelihood that it would have been constructed by Aes Sedai, then restricted to only non-channelers is so slight as to be a ridiculous assumption.
True, such a restriction seems quite silly as it would ban themselves from using it. Men who channel, however, would have been hunted down, gentled, and kept under close watch and guard on general principle. All of the memories are from the Third Age where
Saidin is tainted, after all - this is explicitly confirmed by Mat's own mental recollection at one point. So, men who channel would have been rare and kept away from the
ter'angreal as a coincidental side effect of other policies regarding them. Women who can channel do not appear in Mat's memories for the same reason that women in general don't. Presumably, the Finns decided for some unknown reason to not cross gender boundaries with their memory filling.
Broad strokes or fine, if it is time turning and not just historical paradigms repeating then you are stuck with certain major events repeating closely enogh that the major players are the same. If it is simply that archetypes and paradigms are repeated then you lose the "wheel" and get something linear but cycling.
Select major events, yes, and that would certainly include some battles. Names and physical appearance, though, close enough that Mat would not realize they were different, is a bit to much imo.
And yes, given that they are so selectively non-channy there are huge reasons, already cited that they would have to not be the users of the terangreal. Another big one is the fact that many of the memory tracks include the death of the donor. Kind of hard to use the terangreal AFTER dying. ;P Now it's possible it's a plot hole that the author overlooked, but it's more likely that a lot of mistaken assumptions have made it into FAQs.
No, it's a plot hole that doesn't exist. There are multiple perfectly reasonable explanations and direct explicit authorial confirmation.
Given the restriction of Third Age males, probably due entirely to the Finns choosing to use the most recent available memories of the correct gender, the lack of channelers using the
ter'angreal is no surprise at all. For memories including dying, Mat believes the Finns established some sort of link with him and every other person to ever visit them that allows them to access their memories at any point after the link is made. They also are known to have some ability to see the future - some of the answers they have given prove that - and might be able to browse a person's future memories when he's there. Mat's theory seems more likely to me, though, as I'd expect the Finn's future sight to be less detailed than that.