SPOILERS AHEAD. DO NOT READ ANY FURTHER IF YOU HAVEN'T FINISHED THE GAME!!
The crystal which Garnet wears throughout the game is the Royal Pendant of Alexandria, one of the four Crystals which when combined, can be used to summon the Eidolon Judge Alexander (the other three are Lindblum's Falcon's Claw; Cleyra's Desert Star, and finally Madian Sari's Memory Earring, which Eiko posesses).
Hence the real reason for Brahne's war - it was actually a front for Kuja to steal the Four Crystals so he could summon Alexander and use it in his vindictive battle against Garland. But perhaps I'm getting a little too far ahead of myself.
The reason Garnet left the pendant on the ground where it fell in the final scene, is because of all the heartache and pain the pendant had caused her and her friends and family. To her, it represented the painful past, and as Zidane was waiting on the Prima Vista's stage for her, she had to let go of that past in order for her to face her future and her responsibilties.
However, the Royal Pendant and the crystal in the logo are not the same thing. The game's logo is actually the Crystal of Life; the origin of all life and existence. It appears in the game right at the end of Memoria/Crystal World, where Kuja, aware of his finite life span, threatens to destroy the Crystal, and therefore the universe, along with it, after coming to the insane conclusion that if he is to die, then the rest of the universe has no right to exist without him.
When Kuja threatens the Crystal, it inadvently triggered the arrival of Necron, the Crystal's antithesis if you will. Where the Crystal gives life, Necron destroys it. Accordingly, beating Necron sends it back to whatever dark hell it came from and the Crystal is saved. That is the meaning behind the Crystal in the logo - it symbolises life and existence; the game's main 'theme'.
As for the FMV scenes, I think they can be found at RPGamer.com, but I'm not too sure about that. Otherwise, if you have a reasonable PC, you could always view the FFIX FMV files on your PC direct from the game discs with a Playstation media viewer like PSMplay orPSXmc (which can be found at Zophar's Domain at
http://www.zophar.net/utilities/psxutil.html ).
One word of caution about PSXmc though. Don't use the Shareware version, as it's a 30 day trial which locks up and becomes unusable after the trial period is up. Get the Freeware version instead (both versions can be found via the above link). I use the Freeware version of PSXmc all the time to play FMV movies and certain music files from PSX/PS1 games and it works very well.