Huh?

By the information in New Spring, Galad had a better claim to the Sun Throne than Elayne. By primogeniture, first of all. Only Andor excludes the males from the succession line, not Cairhien. Taringail himself was excluded because of his wedding to a foreign heir, which suggests it's most extraordinary Cairhienin decided to recognize Elayne's claim. Even much better than either Elayne's or Galad's claim is Moiraine's. Neither Galad nor Moiraine have pushed their claim forward, that's what let the door open for Elayne. If not for the desire to be lead by anyone but Rand and the knowledge Elayne is the only one he was ready to accept in this role, for the anarchy in the country and the need for a ruler, Elayne's claim would have been rejected. If a Caihrienin heir could be passed on for both the Throne and the High Seat of House Damodred (which passed to his cousin Barthanes) on the sole basis that he was married to the heir of a foreign throne, imagine where that left Elayne, who is a foreign ruler and an Aes Sedai...

Taringail did not marry into House Mantear. He remained a Damodred, until his second wedding when Morgase forced him to marry into House Trakand. Galad is not part of House Mantear either, merely tied to it by his mother. We don't know if he's officially part of House Trakand - we don't know the legal rules of Morgase's adoption. He retained his name in any case, which the other Andoran lords who marry into another House do not (the lords whose wife lead the House have her name, that's how we know this).

Why the High Seats let Morgase raise Galad? Isn't it obvious: the adopted son of the Queen of Andor had a claim to the Sun Throne, a claim Andor might be able to push forward one day if House Riatin failed. No house would have let any other but Andor's Royal House get their hands on Galad... it might have eventually given that House a power to rival Andor's.

The fact Elayne is AS and a foreign ruler wasn't enough in the current circumstances to reject her claim of the Sun Throne. I don't think the fact Galad leads the WC would be enough to prevent him from ruling in Cairhien.

I'm not saying this is likely, though. I'm just not convinced Elayne will retain both nations - and if she does she'll need help to rule both (Galad's, Moiraine's?), but the way things are going, it sounds like she may loose in the Last Battle the heart of Andor and its capital. Will Mayene stand after TG?

It's not impossible Elayne has to make Ishara's choice, when she doesn't have enough of a population after TG to justify claiming that much territory. No, I don't think Elayne would choose to relocate in Cairhien by choice. But I hardly see her relocating in Baerlon or Whitebridge either... Aringill is already a more logical choice as it stands between her two nations. But she may have to be pragmatic and relocate in her second capital if Caemlyn gets destroyed and Cairhien still stands, and by pragmatism too she may have to gather up her population.

Aviendha's unreliable vision offered very little clues. We know the Court of the Sun name exists (it appears in a Fourth Age quote), but in the vision Cairhien has fallen. Caemlyn/Andor stands, and is apparently part of a pact which name suggests it's an alliance of Perrin-Elayne originally... but weirdly it's the body of Andor's Lion and the head of the Eagle... But considering it's not what will happen, it doesn't tell us much.