You don't need to run Coil synced through Party Finder. You could create party finders for running unsynced to just see the story.

You'll understand what the Binding Coil of Bahamut is, why Bahamut is there and what the dragons are doing there as well when you complete each section and see the related story.

Alisaie is there as she wants to find the truth about what happened to her grandfather. She initially did not see eye to eye with Alphinaud about the way to do things and parts ways with him (you see those scenes when you are doing the envoy tour around the 3 city states before your first dungeon) but later on in MSQ she becomes part of the Scions.