Its a one shot. Unless you saved before the legendary battle or dont mind re-doing Mt.Coronet then dont save and restart your ds
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You have to beat the elite four and then look for him in the same spot he was when he fainted
If you've already fought and beaten Palkia without catching it, then that's it. It does not reappear.
No once any Pokemon has fainted you can not capture it sorry
it already did dummy - where have you been
If you have already caught Palkia or Dialga (Palkia-Pearl Dialga-Diamond) then that means that you have already defeated Team Galactic, and you can go to the Pal Park, try to find more legendaries, or anything you want to do, but if not then you have to go to the place where you will catch either Palkia or Dialga and right before you catch them you will encounter Cyrus, the leader of Team Galactic. If you want to know where to find Palkia or Dialga and encounter Cyrus try looking up something like "where do you catch Palkia or Dialga?" or something like that.