A reread would have helped you there...

There is a line of succession in Cairhien, but in all your arguments you fail to factor in the fact the Game of Houses play there as great a role, if not greater. You also forget these aren't modern states, with an independent judicial branch. Successions are always messy, because the ultimate holder of judicial power is the ruler and when he/she's dead there's no longer an executive judicial power to force the players to respect the laws... laws are respected only if the most powerful players manage to impose it. Situations like this happened in nearly all the European monarchies, and it does in WOT. That's exactly what happened in Andor, when the likes of Arymilla disregarded the succession law. That's exactly what happened in Cairhien with Elayne (at the very least, it's obvious if it's the law a candidate can't be married to a foreign ruler to avoid a foreign throne to control Cairhien, a foreign ruler herself would be even less eligible, by law or by opposition of the Houses).

Even the direct heir needs his own House to support his claim for the Throne (meaning any unsuitable candidate, however high standing, or one that wouldn't be able to gain support from the other Houses, may be pushed aside), and needs wide support from Houses of sufficient power to make the move. Taking the throne of Cairhien is essentially all about being powerful enough that those who would oppose you won't move against you, or if they do, will be defeated or isolated. That's how Galldrian gained the throne, and that's how Colavaere gained the throne. It's similar in Andor, just more formalized and less messy. Arymilla, Nean and co. had nowhere the blood claim of Elayne (not to mention by Andoran law Elayne was the recognized heir - the Houses chose not to support her, turning this into a Succession war), but if they managed to get the support of ten Houses, that would have been overlooked.

Taringail wasn't Laman's heir, Fionwe invented that. By Cairhienin law (it's stated black on white in NS), he couldn't even be Laman's heir because of his wedding to Tigraine (and Morgase later). Laman's heir was his next brother (Aldecain or Moressin, both heirless), then the next. Their brother, Moiraine's father, was dead, and so his son was next, and Moiraine's half-sisters. Then Moiraine, who was youngest child. There were more distant relatives, like Laman's cousin Barthanes, Caraline, probably more people at the time, but clearly they were not yet powerful enough within House Damodred at the time (it's obviously the deaths of Laman, his two brothers, later of Taringail - and Moiraine's self-removal from Cairhienin politics - that opened the way to their rise toward more prominence). Barthanes later on (by the time of TGH, I mean) might have been able to pull it off and regain the Throne if Galldrian died, but clearly Caraline herself wasn't powerful enough to forestall Colavaere.

House Damodred had gained a very bad reputation with Laman and the predecessors (except his mother), which didn't help its prospects with the other Houses. It was also stripped of its three top players in one blow. The moment Moiraine heard of her three uncles' deaths, she figured out the Crown would most likely pass to another House, that Daes'daemar would decide which one...

Taringail was denied a claim by law, because he was married to the Queen of Andor (Taringail must have bitterly regretted having married Morgase not long ago when he heard Laman had died - his frustration at having missed the Throne for that marriage probably played a role in fuelling his later ambitions in Andor, though I still suspect Taringail never truly plotted to murder Morgase. I believe it's Elaida who manipulated Thom into removing Taringail, then moved to remove Thom himself. I think it's Elaida and Thom Taringail plotted to murder or remove. It's them who stood in his way to political influence over Morgase, which he needed to build to one day place Galad on the Cairhienin throne. It's part of the story Moiraine means to reveal to Thom, as it's part of the explanation why his nephew died the way he did).

The law that excluded Taringail was "A stupid Law", by the reckoning of the Grey Ajah/WT that apparently would have seen nothing wrong in the reunion of the Cairhien-Andor crowns, but "a law he (Taringail) can't change unless he was King". Yes, that's one thing you forgot: Morgase was Queen when Laman died, Elaida already her advisor. Obviously, no law stripped Galad of his birthright, why do think Taringail hoped to make him King of Cairhien one day? Why do you think Elayne managed to put her birthright forward? That she succeeded was all a matter of support, but clearly no law stood against her or Galad. Only Taringail was excluded from the line of succession, not his offspring. Taringail excluded by law, Anvaere Damodred was then Laman's closest relative, followed by Innloine, Moiraine, Galad, the sister's children. And then it went to cousins of Laman, like Barthanes, and eventually children of cousins (likely) like Caraline. If Anvaere's children survived, they're clearly not all that powerful.

Moiraine said neither of her sisters would ever manage to gain enough support for the Throne from within House Damodred itself, let alone with the other Houses. Herself wouldn't put her claim forward (and thought as Aes Sedai, that might lead to much opposition, and woes if with WT support she won). Evidently, a Damodred toddler wouldn't be able to gain support either (if that's even legal in Cairhien for a minor to inherit the Throne. Anvaere's children were not even mentionned as possibilities so...), and if Anvaere and Innloine were not suitable for the throne, neither would have been suitable as Galad's regent either anyway.

With the disarray in House Damodred, Galldrian su Riatin managed to find enough support to gain the Throne. He had no direct heir (Toram seems to be a cousin, himself had no direct heir, but a sister). Evidently, Taringail had hopes (were they realistic ones, we'll never know...) he might manage to gain enough support in House Damodred for his son when he was on age in the eventuality Galldrian died (and that's Cairhien... nothing says Taringail intended to wait for Galldrian's natural death) and Barthanes had managed to grab the High Seat of Damodred (we don't know exactly when, but likely quite before Taringail was murdered, as it was some years after Laman's death).