When Bones holds up the evidence bag and says that ballistics had matched the bullet to the gun and that the bullet was found in Beaver's corpse, the bag contains an expended shell casing, not a bullet.
In the opening scene after the mascot costume was shot you see the costume tear open revealing a body that is bent at the waist (as if touching toes) in the bottom of the costume.
But when bones starts to examine the body you can see parts of the costumes headpiece are infused to the head of the victim (which was not wearing the headpiece when it fell out of the costume)
Twice in the opening minutes of the show, characters refer to a full-sized muzzle-loading cannon (mounted on spoked wooden wheels) as a "blunderbuss," which was an early hand-held firearm usually loaded with shot.
Seeley states that Jared has received a dishonorable discharge. Military officers cannot receive a dishonorable discharge. If they are convicted in a court-marital, they can receive a dismissal from service.
When the victim was first identified, Brennan says his age is 23 years old, but later when Booth and Brennan are investigating the victim's room, she says his birthday was 5-11-89, which would have made him 19-20 at the time of his death, not 23.