After chasing Diana into the water, when returning to shore, the dog is chewing a ball in his mouth.
When Diana jumped into the water she was holding a bottle with the note. When swimming to the boat both hands are visible without the bottle. When she gets to the boat, she hands the bottle to Perry.
When Perry is doing the crime reenactment in court with the dog Fritz, he has one of his witnesses Simmons point a pistol at the different attendees as Perry calls them up one by one. When Fritz jumps and grabs Simmons' arm the pistol goes off. They would not use a real pistol, even one possibly loaded with blanks, for the demonstration.
When Perry and Paul demonstrate where the missing bullet may be in the ceiling for Lt. Drumm, the bullet is lodged in the ceiling at a perfect 90 degree angle. However, if fired from across the room like Perry demonstrated, it would have entered the ceiling on an angle of about 50 degrees. Plus the shooter would have to be located across the room directly under where the bullet was in the ceiling and firing straight above his head for the bullet to lodge as Drumm found it.
On the night of the murder, Paul Drake is standing just outside the gate of Addison Powell's estate. When Henry Simmons comes out, he speaks to Paul and addresses him by name ("Mr Drake"). But at that point in the story, they had not met, so he wouldn't know his name.