I think TinyRedLeaf addressed the questions I would have replied to quite well, already, but...
It's hard to say for sure how much of Her weakness can be attributed to bits of the Light being on other worlds and how much is just that seven straight Calamities (and seven sets of Warriors of Light that can harness Her strength) have really taken their toll. Hell, even empowering us, casting out Lahabrea, and the Ultima shields cost Her pretty dearly.
As far as we know, Minfilia is the only Emissary - The First may have been the only time that Ascian machination we defeated so hard that a gap was placed between a flood of Light and a Rejoining. The Warriors of Light may have just done everything they could have in the past and kept ultimately losing ground; there might not have been a flood of Light to stop. But this is all speculation - it could all be wrong.
Technically ... There's Lighting / Fire / Earth / Ice / Water / Wind. But there are also crystals of Darkness, and since 1.0 there have been manifestations of imbalanced aspects that tip towards two adjacent elements, even three when it's the "Astral" trio (Wind / Lighting \ Fire) or the "Umbral" trio (Water \ Ice / Earth). The entire Chrysalis, for example, is made of Dark Crystal - from the door to the platforms to the statue. That, combined with the fact that it looks like the inside of the Burning Heart, has me suspicious that the interdimensional rift is actually within the moon - perhaps within the "Fathercrystal" itself.
You know me; I hold logic and speculation pretty dear until we get to issues of "keeping the faith". I'll support doubt, I'll back interpretations to the contrary, but it just seems like a more powerful adventure if you hold fast to hope. Most tangible evidence so far points to the Twelve being seeds of faith that descend from legends of the Archons and the Warriors of Light, every era reinforced and strengthened when history repeats itself. The Circle of Knowing were twelve figures bringing accurate warnings of the future and even today the commonfolk will be quick to say, "THE ANGELS HAVE RETURNED TO US!"
Still ... I keep the faith that the Twelve are something. My current baseless headcanon is that they existed in an unwritten era of Prehistory between the division of the Source and the first Rejoining; that they were extremely powerful forms of Life that existed early on as a means to lead and guide, but ultimately it didn't work out. But who knows. Maybe they were all dragons, eh!?
Correct. Louisoix tried to tap into a summoning ritual to essentially summon primal versions of the Twelve and direct their strength towards sending Dalamud back into the heavens before ultimately killing himself to prevent their corporeal manifestation (Bahamut broke the spell before it got that far). From the symbols in the sky to the literal recreation of Dalamud around Bahamut, it's strongly suggested that these are just manifestations of faith and aether.
What the Twelve are otherwise - if anything - is still a great mystery.