So, spoiler alert:
1. We go into the Binding Coil to prevent primal Bahamut from reawakening and annihilating Eorzea.
2. Rewatch the opening cinematic to A Realm Reborn. Bahamut sends shards of Dalamud everywhere.
3. Yes, the binding coils are parts of Dalamud that bound primal Bahamut in place.
4. Bahamut was originally one of Midgardsormr's 7 children, who went with Tiamat to Meracydia, which was eventually attacked by the Allagans during their height. Bahamut was slain, and then Ascians convinced Tiamat and their brood to resummon him as a primal.
5. During the Final Coil Turn 4 you are fighting Bahamut Prime, a smaller core essence of the primal form of Bahamut buried in his heart.
6. See #1. We save the planet.
Also, there are several lore posts about this in much more depth floating around, and if you go do the Coil series and watch cutscenes/read dialogue, as well as do the Heavensward MSQ you will get all this information and more.