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I think for the Templar outfit you can start earning keys after you finish sequence four. As for the Mayan outfit I think it's around sequence 5 or 6 you first see where you put the pieces. Keep in mind the missions and mayan stelas are on many different islands some are in hard sections of the map. So you may have to wait to get some of the parts till you get farther in the game.
Not very far, the story doesn't really hinder you from collecting them. It's some of the higher level ships that guard some of the locales they are located on. You can try stealth swimming but they will snipe you with the swivels.
I'm glad there are outfits sans hoods... It made sense with AC and Altair, kinda with Ezio as he inherited his father's, but it's getting to be a bit of a running joke.
One templar says to the other, "How can you tell there's an assassin around? For starters, there the only ones wearing a damn hoodie."
Which is why I use hoodless attires with Edward, and hope Ubi feature them going forward. In eras or settings where they weren't commonplace it looks ridiculous... You don't see Templars walking around with crosses or the Abstergo logo on their attires, aside from the first game.