In the movie why did the Phantom take Christine's ring, then give it back? Why did the Phantom leave it on her grave when she gave it to him so he could remember her with something she personally owned?

The Phantom putting Christine's ring on her grave is extremely symbolic, like a lot of things in the 2004 movie. There are many ways of interpreting it, but the way I see it, it's that finally, after 50 years, the Phantom has finally accepted that Christine loved Raoul and not him. He then arranges with Meg (and not Mme Giry, who would be in her nineties) that Raoul buys the musique box to take it to Christine's grave, so he can see the ring with the red rose. The rose coloring itself as the end symbolizes forgiveness and peace: those are the most powerful proofs of l’amour of all.