2003 Daihatsu Cuore Radiator water leakage

When I fill the box near radiator with water the water goes down instantly into the radiator and the same is leaking from the cover cap of the radiator while its tightly closed.

If radiator is full then why the water is going down and if it's not then why won't it take it instead of leaving it from cover cap.

When car is cooled down all night in garage and I come to start it in the morning, even though the battery is new and its not the battery issue as I purchased a new one but the same issue was with and before new battery. What happens is that car won't start unless I have someone outside opening the bonnet or hoof and tapping his hand against the air filter pipe there and car starts in first try.