As I understand, the F number is a ratio of the focal length to the diameter of the aperture. It doesn't tell you how much light gets through. You could have two lenses with the same max aperture but one could have a much bigger front element that sends more light through it.
No the F number is the same across sensor sizes
at F2.0 on a compact and F2.0 on a DSLR the exposure are supposed to be the same as long as the ISO and shutter speed are also the same.
larger sensors will introduce less noise at a given ISO but exposure levels are supposed to be equal
EDIT didn't see slashbinslashbin's reply so you can ignore this