What is both possible and impossible at the same time?

A car is crossing a 20 km-long bridge. The bridge can support at most 1500kg of weight over it. If somehow, the weight on the bridge becomes more than that, it will break.

Now, the weight of the car is exactly 1500kg. At the midway, a bird comes and sits on the roof of the car. This bird weighs exactly 200 grams.
Can you tell if the bridge breaks at this point or not?

Forwards I am heavy, backward I am not. What am I?

He runs but never walks. Murmurs, but never talks. Has a bed, but never sleeps. And has a mouth, but never eats?

There are two dice with empty faces in front of you and a marker. You can mark any number on each of the faces of the two dice, but you have to display all 31 days of the month using the two of them.

Which numbers will you mark on which dice so that you can easily depict all the dates of the month?

They do not possess flesh or feathers or scales or bone
But still, they are blessed with fingers and thumbs of their own.

They came out at night, without being called.
They lost at day, without being stolen.
Who are they?

There is something which you can be arrested for attempting but not for successfully completing.

John was so frustrated to be the suspect of being a sleeper terrorist that he shoot himself right between the eyes with a revolver in his toilet. The revolver and the bullets were accurate. Minutes later John walks out of the toilet unharmed.
How can this be possible?

Can you find out the smallest number that can be conveyed as the sum of three squares in three unique ways?