Why do they separate some parts of PCBs through making wide holes?
It's for better high-voltage safety. 
Unrelated, because it's not noise. 
Very. Air is a much better insulator, and it doesn't get wet/have a conductive film form. 
Basically, a circuit board can get dirty and/or wet. This makes the surface of the board slightly conductive, and once it's a bit conductive, electrochemical action can make it very conductive, and then you get a free fire.
Isolation slots / creepage distance cutouts/air gaps.