Paul's mother is surprised when he gets up at 4:00 in the morning at the beginning of the movie. Later on, she tells Mathewson that Paul has been getting up this early - and earlier -regularly for years.
Plutonium must be alloyed with another metal (usually gallium) in order to prevent forming allotropes which cause it to crack while cooling. Cracks in the pit would have significant impact in the weapon, and could result in a fizzle (non-nuclear explosion.)
During the process of making the sphere of plutonium Paul doesn't use any type of protection (filter) from airborne plutonium particles. There is a high probability of death when plutonium is inhaled. As a minimum it results in radiation sickness.
Paul shows Jenny a fistful of four leaf clovers that he had picked some time ago from the grounds of the lab. When revealed, they're crisp and freshly picked. If they were in his pocket for as many hours, they would have withered and been much darker in color.