RX100 - Q on image size and sensor
The entire sensor is used yes, the camera does a downscaling before writing the jpeg.
As for noise, if you compare the 10mp image at 100% with the 20mp image at 100% then yes. But if you view them at the same size ie. either printing them at the same size or just viewing both at say 2000 pixels wide it's not going to make a difference.
Shooting for fun and memories.