when you turn on the key and do not start the car, all lights are supposed to glow up indicating that all systems are receiving current. when you start the car, only those lights keep glowing which are under work i.e. blue temp light stays up as long as engine has attained its operating temperature. just like hand brake sign, it turns on when you have pulled the hand brake and goes off once you release it. regarding you queries, you can let you car started for a minute or two in the morning, no big deal, but you can start driving right away with no issues. this is, when you keep you car idle on cold, it will take more time to attain operating temp when compared time it will take if you start driving right away. the negative about keeping you car started on cold is at cold, car's ECU will be in open loop and will be feeding more fuel to engine, that means more carbon buildup in combustion chamber. the quicker your car gets to operating temperature, the better.
