No, the entire world will not be dark. There is always some part of the world that is exposed to the sun. If it is night in North America, then it is day in Europe. And vice versa.
No, turning off all lights will not stop global warming. It may conserve energy usage, but is still not enough to stop global warming. Because global warming comes from driving cars, methane from cattle and cows (because we eat too much beef), and from industrial output (such as factories), and coal-burning industrial plants.