Ok. For Anima, all you need is the destruction spheres from all the temples. If you didn't get the Bevelle one when you had the chance, tough luck, cause now you can't get Anima, which means you can't get the Sisteres.
I'll adress your questions in order.
1 Zanarkand Puzzle: Go back to the cloister in Zan. and step on all of the white squares. it'll give you a destruction sphere, and insert that in a slot somewhere.
2 Celestial Mirror: Race the chocobos, win, and get the cloudy mirror at the Remiem Temple. Then, go into macalania forest, in the entrance where the save spot is. I believe if you talk to the people there, then go to the place where you all rested after Bevelle and talk to the person there, then go back to the save spot area, there will be a shiny path, go up the shiny path on the left, talk to the kid, go back down, talk to the parents and the kid, go back up to where the kid was, and touch the tree thing, and it'll change your cloudy mirror into the celestial mirror.
3 Omega Weapons: They're actually called the Ultimate Weapons, I believe