Yes they do. However during the warm up period a petrol engine will add more fuel. When the temperature is low then this takes longer. For a diesel the injection timing advances but no more fuel is used during cold weather. Diesel does take longer to warm up regardless of temperature.
You sure about that. My diesel cars have been and are much less fuel efficient until they are up to temperature. After all the oils are thicker and theres more friction when they are cold for a start.