I've never seen a cooling system have pressure in it after sitting over night and the motor has not been started. I have seen motors with bad head gaskets pressurize if started. The motor does not need to be at operating temperature to pressurize if the head gasket is leaking compression into the cooling system. If you really do have pressure in the morning after sitting over night I would have to think you have a bad head gasket and a bad radiator cap that is not releasing the pressure which is high enough the system still has pressure in the morning. That's not normal.

The system is always pressurized. It shows you do not have a leak. You should not open the radiator cap at all. Check the antifreeze level in the overflow reservoir. It should have a cold line and a hot line at different levels to show how much antifreeze there is.

Absolutely! If there is no pressure the cooling system has a leak even after sitting 12 hours. If you don't believe it, ask the dealer of your car or any trusted private mechanic. If there is no cooling system pressure, the coolant recovery system would not work!

? is a an untrusted non-mechanic

No the only time it is under pressure is when it is hot

I suggest you use Google to locate information as to the cooling systems of vehicles. That shall give you the basic knowledge how this system works. What were the cooling system issue you were having?

After problems with my cooling system, that morning i decided to check the antifreeze level in the radiator. Before start the car i opened the cap which was under pressure. It sounded like opening a bottle of soda but not so strong. Should the car keep this pressure when its cold or there is some problem ?