The camera should be able to charge any of the NP batteries. There were a couple threads here back when the FX1 was new where people had blown something in the camera by plugging it into the AC line. I suspect this is relatively rare though.
I have a Z1 which came with an external charger. The Z1 does not have a connection for the external power supply like the FX1, instead it has a "fake battery" on a cord that can be attached to the external charger. Personally I never use this and only run my camera on battery power, just to play it safe...