The on-board noise reduction that goes into producing the JPEG may also contribute to the file size as, typically, less noise means smaller file size (more efficient compression).
WRT your "gray background" question - you are referring to White Balance. That is, you need to tell the camera what kind of light you are shooting in to make sure it renders neutrals (grays) neutrally. If a neutral looks too blue, you have your WB set incorrectly - it needs to be more orange/amber/yellow. You don't need to adjust individual colors, just set the white balance properly by choosing the closest preset WB on the camera (Daylight, Tungsten, Shade, Fluorescent, etc.) or use a custom white balance by taking a picture of something you know is neutral and then setting that image as your Custom WB - read the manual to see how you do this. It is very straightforward and takes a few seconds once you get the hang of it.