She doesn't lip sync. Sure there's still some pre-recorded vocals being played in the background and she sings on top of them (hell, Jon has Richie's voice filling in for the high notes, Gaga has her own. Axl Rose too.) but absolutely no one can dance like she does and sing perfectly at the same time. Even Michael Jackson couldn't. But of course when she's at the piano, she's 100% live.
(singing starts at the 2:00 mark)
As for the songwriting, if you like Fast Cars better than Paparazzi then that's your opinion. It's certainly not mine.
Last but not least, Jon seems to pride himself with singing live but he's not exactly been really good at it since 1996 so...