Some personal analysis in a Q&A format...
Why seal Bahamut? Why not just kill him?
Because a defeated primal can be re-summoned from the aether. We've experienced this problem first-hand. By sealing Bahamut, you lock up his sentience and prevent that being drawn back out into a new corpearal form by the dragons.
Why would ancient Allag banish this "great sin" to the sky?
Because primals absorb aether from the land as food. Move Bahamut's prison far enough away from the land and you are essentially starving him of the energy needed to break free.
((Extra context: Gaius tells us that it was the Allagan Empire that originally banished Dalamud. This suggests it happened during the 4th or 5th Umbral era, since Allag was founded in the 3rd Astral and the 6th Umbral was associated with flooding.))
So how did Darnus bring Dalamud down?
By feeding him. The lunar transmitter looks like a powerful aether cannon, sending aether gathered from the land right up to a hungry dragon. OmNomNom!
A hungry beast will move closer to its food source. And the best source with which to generate this food is Silvertear Falls, both the heart of the aether stream and the heart of Nael's enemy's territory. How convenient!
What are the consequences of Dalamud getting closer?
Bahamut's power and influence begins to recover. This allows a little investment in Darnus to replace a broken lunar transmitter and the ability to pick up the latent aether of other dispelled primals. (So on the point of knocking out primals that feed on the land's aether, it becomes a damned-if-you-do-damned-if-you-don't situation.) By the time you've properly dealt with Darnus, Dalamud is close enough for it to not matter any more. Sure that extra energy feed would have been nice, but Bahamut can take care of himself by now.
Next topic.
Do the Twelve exist?
Yes. During the third Gridanian quest, "Beckon of the Elementals", a moogle brings you a message.
"The great arbor spirit Everschade has spoken and we have listened, kupo. Those beckoned by the forest are messengers of Nophica herself."
There is plenty of other anecdotal evidence that others have shared.
So how are they summoned?
Louisoix appears to do so through the use of divine symbols. Likewise, we see the Amaljaa's summoning of Ifrit in a cutscene involve placing a symbol on the ground before Ifrit comes forth. This theory is reinforced by information that the arcanist class will form the basis of the summoner job in ARR.
So is Louisoix an arcanist?
We can probably assume so and likewise for Urianger, who wields a type of staff seen used by the arcanists of Maelvan's Gate. This also helps explain the presence of a divine symbol upon Louisoix's staff and his admission that it was by his hand the thirteen "altars" had appeared (this admission is made at the start of "Living on a Prayer").
So why the pilgrimage? Why not just cast the spell?
An aetherial being can hardly be summoned unless its sentience is roused. It needs to WANT to come out (and Ifrit and Garuda are all too happy). The Twelve were in a deep slumber, possibly acutely aware of what impact their corpearal presence would have on a realm that has had a good chunk of it's aether consumed in a giant seal. We needed to get through to them that we needed them again. Badly.
And who better to lead those prayers than the hero who defeated Darnus whose exploits have been spread and praised throughout the grand companies? You have become a leader and a role model and will be followed and imitated for maximum effectiveness. You are a beacon to the people more than the twelve directly.
Why would the Empire allow this?
They wouldn't. Obsessed with Nhilism, Van Darnus has been witholding the true nature of Dalamud and deceiving His Radiance.